Invited Talks
| First Name | Last Name | Title: | Abstract: |
|---|---|---|---|
| Ricardo | Alonso | On the dynamics of dissipative particles | In this talk we bring two examples of dissipative particle dynamics: inelastic particles, which are associated to dilute granular flows, and reacting particles, associated to ballistic annihilation. Extensive numerical simulations show that suitable versions of the Boltzmann equation are reliable models for such interesting dynamics. We explain the models and explore different analytical methods that, applied to the Boltzmann equation, allow studying different aspects of dissipative particle dynamics such as precise density tails, optimal rates of dissipation, and convergence towards non-Gaussian equilibrium. One of the key ingredients is the use of self-similarity to regain conservation laws that are lost due to the dissipation process. Some interesting problems in this area are mentioned at the end. |
| Myoungjean | Bae | Detached shock past a blunt body | It is well known that when a steady supersonic flow with uniform data impinges onto a two dimensional symmetric wedge whose half-angle is less than the detachment angle, then there are two possible steady configurations: the steady weak shock solution and strong shock solution. In both cases, the shocks are attached to the vertex of the wedge. Many previous studies show that both weak shock solution and strong shock solution are structurally stable if the boundary of the wedge is slightly perturbed away from the vertex. However, if the tip of the wedge is smoothed out so that the wedge becomes a blunt body, then it is impossible for a shock to be attached to the tip of the wedge. In this talk, I present the existence of detached shock solution past a blunt body. This talk is based on a joint work with Wei Xiang(City University of Hong Kong). |
| Lydia | Bieri | The Einstein Equations and Gravitational Waves | In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear pde has served as a playground for all kinds of new problems and methods in pde analysis and geometry. A major goal in the study of these equations is to investigate the analytic properties and geometries of the solution spacetimes. In particular, fluctuations of the curvature of the spacetime, known as gravitational waves, have been a highly active research topic. In 2015, gravitational waves were observed for the first time by Advanced LIGO (and several times since then). Understanding gravitational radiation is tightly interwoven with the study of the Cauchy problem in GR. I will talk about geometric-analytic results on gravitational radiation and the memory effect of gravitational waves. We will connect the mathematical findings to experiments. I will also address recent work with David Garfinkle on gravitational radiation in asymptotically flat as well as cosmological spacetimes. |
| Rémi | Carles | Rigidity in generalized isothermal fluids | We investigate the long-time behavior of solutions to the isothermal Euler, Korteweg or quantum Navier-Stokes equations. By writing the system with a suitable time-dependent scaling we prove that the densities of global solutions display universal dispersion rate and asymptotic profile. This result applies to weak solutions defined in an appropriate way. Compactness of bounded sets of such weak solutions is obtained by introducing modified entropies adapted to the new unknown functions. This is based on a joint work with Kleber Carrapatoso and Matthieu Hillairet. |
| Christophe | Chalons | On all-regime and well-balanced Lagrange-Projection schemes for compressible fluid systems | It is the purpose of this talk to provide an overview on recent advances on the development of Lagrange Projection like numerical schemes for compressible fluids systems with source terms. The key idea of the Lagrange-Projection strategy is to decouple the acoustic and transport phenomenon. When combined with a Suliciu like relaxation technique, the Lagrange-Projection strategy leads to efficient implicit-explicit discretisations on fixed unstructured grids, with CFL conditions driven by the (slow) material waves and not by the (fast) acoustic waves. The resulting scheme also satisfies a fully discrete entropy inequality. As we will see, the strategy is very well-suited to design efficient all-regime and well-balanced numerical schemes. For the purpose of illustration, we will first consider the nearly incompressible limit of low Mach number flows and the diffusive limit of the gas dynamics equations with source terms, for which asymptotic-preserving schemes are proposed. We will also show that the strategy allows to design fully well-balanced schemes for the shallow water equations. By fully well-balanced, we mean here that the scheme is able to preserve stationary states with non-zero velocity. |
| Alina | Chertock | Structure Preserving Numerical Methods for Hyperbolic Systems of Conservation and Balance Laws | Many physical models, while quite different in nature, can be described by nonlinear hyperbolic systems of conservation and balance laws. The main source of difficulties one comes across when numerically solving these systems is lack of smoothness as solutions of hyperbolic conservation/balance laws may develop very complicated nonlinear wave structures including shocks, rarefaction waves and contact discontinuities. The level of complexity may increase even further when solutions of the hyperbolic system reveal a multiscale character and/or the system includes additional terms such as friction terms, geometrical terms, nonconservative products, etc., which are needed to be taken into account in order to achieve a proper description of the studied physical phenomena. In such cases, it is extremely important to design a numerical method that is not only consistent with the given PDEs, but also preserves certain structural and asymptotic properties of the underlying problem at the discrete level. While a variety of numerical methods for such models have been successfully developed, there are still many open problems, for which the derivation of reliable high-resolution numerical methods still remains to be an extremely challenging task. In this talk, I will discuss recent advances in the development of two classes of structure preserving numerical methods for nonlinear hyperbolic systems of conservation and balance laws. In particular, I will present (i) well-balanced and positivity preserving numerical schemes, that is, the methods which are capable of exactly preserving some steady-state solutions as well as maintaining the positivity of the numerical quantities when it is required by the physical application, and (ii) asymptotic preserving schemes, which provide accurate and efficient numerical solutions in certain stiff and/or asymptotic regimes of physical interest. |
| Young-Pil | Choi | Asymptotic analysis of Vlasov equation with nonlocal forces | Collective coordinated motion of autonomous self-propelled agents with self-organization into robust patterns appears in many applications. Apart from its biological and evolutionary relevance, collective phenomena play a prominent role in many other scientific disciplines, such as robotics, control theory, economics and social sciences. In this talk, we will consider kinetic type models describing the collective behaviors and discuss a quantitative estimate of a large friction limit to a continuity equation with nonlocal velocity fields, called aggregation equations. |
| Sara | Daneri | Non-uniqueness for the incompressible Euler equations up to Onsager's critical exponent | Our work is related to Onsager's conjecture, proven by Isett and refined by Buckmaster, De Lellis, Szekelyhidi and Vicol, according to which below Hölder regularity 1/3 there exist solutions of the incompressible Euler equations which dissipate the total kinetic energy of the flow. In a series of papers we deal with the associated initial value problem, with regularity below Onsager's critical exponent. Although in a smooth setting dissipation of the total kinetic energy implies uniqueness of solutions with the same initial datum, we are able to show that there exists a dense set of Hölder 1/3-\eps initial data, each admitting infinitely many Hölder 1/3 dissipative solutions of the Euler equations. In particular, an instance of the h-principle holds for the incompressible Euler equations below the critical exponent, similarly to what happens for C^1 isometric embeddings of Riemannian manifolds in R^n . Moreover, in order to prove the density in L^2 of such wild initial data we introduce a family of stationary solutions of the Euler equations, the so called Mikado flows, which proved to be the crucial ingredient in Isett's proof of the Onsager's conjecture. Some of the results are a joint work with E. Runa and L. Szekelyhidi. |
| Michael | Dumbser | High order ADER schemes and symmetric hyperbolic models for Newtonian and general relativistic continuum physics | We present a new family of high order accurate ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiters for the solution of nonlinear systems of hyperbolic partial differential equations than may also contain nonconservative products and stiff source terms. The new schemes can be used on space-time adaptive Cartesian meshes (AMR), as well as on fixed and moving unstructured simplex meshes. Subsequently, a new unified first order symmetric hyperbolic and thermodynamically compatible (SHTC) model for the description of classical Newtonian continuum physics is presented. The model is based on previous work of Godunov, Peshkov and Romenski (so-called GPR model) and allows to describe nonlinear elasto-plastic solids as well as viscous and inviscid Newtonian fluids coupled with electro-magnetic fields within on single set of governing equations. The main results of a formal asymptotic expansion of the model in the stiff relaxation limit are summarized. Next, the GPR model is extended to the framework of general relativity, where it describes the dynamics of relativistic elastic solids as well as ideal and non-ideal (viscous) relativistic fluids. As in the Newtonian case, a formal asymptotic expansion reveals again the structure of the viscous stress tensor in the stiff relaxation limit. The new model is symmetric hyperbolic and thermodynamically compatible; it is covariant and causal, hence it has bounded signal speeds for all involved processes, including dissipative ones. We present numerical results obtained with high order ADER schemes for inviscid and viscous Newtonian and general relativistic fluids, as well as for Newtonian and general relativistic solid mechanics. At the end of the talk we propose a new strongly hyperbolic first order reduction of the CCZ4 formulation of the Einstein field equations of general relativity and its solution with high order ADER discontinuous Galerkin finite element schemes. References: [1] S.K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR 139:521?523, 1961 [2] S.K. Godunov, E.I. Romenski, Nonstationary equations of nonlinear elasticity theory in Eulerian coordinates, J. Appl. Mech. Tech. Phys. 13:868?884, 1972 [3] E.I. Romenski, Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics, Math. Comput. Model. 28:115?130, 1998 [4] I. Peshkov, E. Romenski, A hyperbolic model for viscous Newtonian flows, Contin. Mech. Thermodyn. 28:85?104, 2016 [5] M. Dumbser, I. Peshkov, E. Romenski and O. Zanotti. High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: viscous heat-conducting fluids and elastic solids. Journal of Computational Physics 314:824?862, 2016 [6] M. Dumbser, I. Peshkov, E. Romenski and O. Zanotti. High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics. Journal of Computational Physics. 348:298?342, 2017 [7] M. Dumbser, O. Zanotti, R. Loub?re and S. Diot. A Posteriori Subcell Limiting of the Discontinuous Galerkin Finite Element Method for Hyperbolic Conservation Laws. Journal of Computational Physics, 278:47?75, 2014 [8] W. Boscheri and M. Dumbser. Arbitrary-Lagrangian-Eulerian discontinuous Galerkin schemes with a posteriori subcell finite volume limiting on moving unstructured meshes. Journal of Computational Physics, 346:449-479, 2017 [9] M. Dumbser, F. Guercilena, S. K?ppel, L. Rezzolla and O. Zanotti. Conformal and covariant Z4 formulation of the Einstein equations: strongly hyperbolic first-order reduction and solution with discontinuous Galerkin schemes. Physical Review D. in press. https://arxiv.org/abs/1707.09910 |
| Mikhail | Feldman | Uniqueness and stability for shock reflection problem | We discuss shock reflection problem for compressible gas dynamics, von Neumann conjectures on transition between regular and Mach reflections, and existence of regular reflection solutions for potential flow equation. Then we will talk about recent results on uniqueness and stability of regular reflection solutions for potential flow equation in a natural class of self-similar solutions. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, and prove uniqueness by a version of method of continuity. A property of solutions important for the proof of uniqueness is convexity of the free boundary. We also describe a result on instability of regular reflection solutions for potential flow with respect to the non-symmetric perturbations, and discuss that for Euler system stability is expected for this class of perturbations. This talk is based on joint works with G.-Q. Chen and W. Xiang, and with J. Hu. |
| Enrique Domingo | Fernandez Nieto | From submarine avalanches models to bedload sediment transport simulations | The main objective of this talk is to present a depth-averaged model that can be used to study submarine avalanches and bedload sediment transport problems, as soon as a finite volume solver to approximate the system. Firstly, an overview on several difficulties related to submarine avalanches modelling will be presented. Namely, the choice of the coordinate system and the effect of the pore fluid pressure on the submerged granular layer. Secondly, a two-layer shallow water type model is presented, which can be considered to study submarined avalanches and bedload sediment transport, that is, for strong and weak interactions between the fluid and the sediment. It can be seen that the two-layer model converges to a Saint-Venant-Exner system when the ratio between the hydrodynamic and morphodynamic time scales is small. Finally, a finite volume method will be presented, highlighting the main difficulties realted to these problems. Numerical tests, by comparing with analytical solutions and laboratory data will be finally presented. |
| Thierry | Gallay | Stability of vortices in ideal fluids: the legacy of Kelvin and Rayleigh. | The mathematical theory of hydrodynamic stability started in the middle of the 19th century with the study of model examples, such as parallel flows, vortex rings, and surfaces of discontinuity. We focus here on the equally interesting case of columnar vortices, which are axisymmetric stationary flows where the velocity field only depends on the distance to the symmetry axis and has no component in the axial direction. The stability of such flows was first investigated by Kelvin in 1880, for some particular velocity profiles, and the problem benefited from important contributions by Rayleigh in 1880 and 1917. Despite further progress in the 20th century, the only rigorous results available so far are necessary conditions for instability under either two-dimensional or axisymmetric perturbations. The purpose of this talk is to present a recent work in collaboration with D. Smets (Paris), where we prove under mild assumptions that columnar vortices are spectrally stable with respect to general three-dimensional perturbations. The proof relies on a homotopy argument, which allows us to restrict the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis in the complex plane. |
| Simone | Goettlich | Non-local conservation laws for material flow problems | The material flow problems under consideration have a granular like structure and allow for a multi-scale model hierarchy. Starting from a detailed microscopic model based on Newton type dynamics, a corresponding macroscopic model is derived, leading to conservation laws with non-local interaction term. Both modeling approaches are fitted against real data from an experimental setup. In addtion to numerical simulation results and theoretical investigations, we address questions of optimal control and extensions to networks. |
| Yan | Guo | On Euler-Poisson System | The Euler-Poisson system describes important physical models for a plasma or a self-gravitating star. We will review recent progresses in the PDE study of the Euler-Poisson system. |
| Yoshiyuki | Kagei | Bifurcation of the compressible Taylor vortex | The Couette-Taylor problem, a flow between two concentric rotating cylinders, has been widely studied as a good subject of the study of pattern formation and transition to turbulence. Consider the case where the inner cylinder is rotating with uniform speed and the outer one is at rest. If the rotating speed is sufficiently small, a laminar flow (Couette flow) is stable. When the rotating speed increases, beyond a certain value of the rotating speed, a vortex flow pattern (Taylor vortex) appears. For viscous incompressible fluids, the occurrence of the Taylor vortex was shown to solve a bifurcation problem for the incompressible Navier-Stokes equations. In this talk, this problem will be considered for viscous compressible fluids. The spectrum of the linearized operator around the Couette flow is investigated and the bifurcation of the compressible Taylor vortex is proved when the Mach number is sufficiently small. It is also proved that the compressible Taylor vortex converges to the incompressible one when the Mach number tends to zero. This talk is based on a joint work with Prof. Takaaki Nishida (Kyoto University) and Ms. Yuka Teramoto (Kyushu University). |
| Zhen | Lei | Strong Null Condition and Global Well-posedness for Elastodynamics | In our recent work, the concept of strong null condition is introduced and incompressible elastic system is shown to inherently satisfy this condition after surgery. We use it to prove the global well-posedness of solutions to the 2D incompressible elastodynamics. The method can be used for the most remarkable progresses on the theory of global well-posedness of quasi-linear wave and elastic equations, which will also be sketched. |
| Knut-Andreas | Lie | Why developing high-resolution schemes for reservoir simulation is challenging | Equations modelling multiphase/multicomponent flow in porous media are widely studied by the hyperbolic community, e.g., as examples of problems with nonconvex and spatially-dependent flux. Various types of high-resolution methods have been developed to solve these equations, but few have managed to permeate into industry use. This is perhaps surprising, given that the numerically predicted outcome of critical engineering decisions can be severely impacted by numerical diffusion and insufficient accuracy. In this talk, I explain some of the challenges and opportunities we have encountered when developing high-resolution methods capable of simulating industry standard reservoir models: 1) Reservoir rocks are typically represented by grids with unstructured topology and polyhedral cell geometries and large aspect ratios. Cells have non-matching interfaces, degenerate geometries, and large variations in sizes. Most methods in the literature are developed for structured topologies or simplices with close to unit aspect ratios. 2) The flow equations are parabolic with a mixed elliptic-hyperbolic character. Strong coupling and large variations in time constants imply that implicit discretizations are necessary also for the hyperbolic part of the problem. Most methods in the literature are explicit. |
| Hailiang | Liu | ON STRUCTURE-PRESERVING HIGH ORDER DG METHODS FOR CONSERVATION LAWS | Systems of conservation laws for field quantities arise in diverse applications. Their solutions may be visualized as evolving observables or propagating waves. When the system is nonlinear, solution profiles can become steeper as shocks or even concentrated as measures, propagation of these profiles cause mathematical and numerical challenges in solving systems of conservation laws. This talk is concerned with the numerical aspects. I shall show how some intrinsic solution properties can be built into the high order numerical approximations through the study of two model classes, diffusion with interaction potentials [1], and multi-dimensional systems of hyperbolic conservation laws [2]. For the former, the three desired solution properties are mass conservation, non-negativity, and the entropy dissipation law. For solving time-dependent Poisson- Nernst-Planck systems, we design an arbitrary-order free energy satisfying dis- continuous Galerkin (DG) method. Both the semi-discrete and fully discrete DG methods are shown to satisfy the corresponding discrete free energy dissipation law for positive numerical solutions. Positivities of numerical solutions are enforced by an accuracy-preserving limiter in reference to positive cell averages. For multi-dimensional hyperbolic conservation law systems we introduce an invariant- region-preserving (IRP) limiter, if the system admits a global invariant region which is a convex set in the phase space. It is shown that the order of approximation ac- curacy is not destroyed by the IRP limiter, provided the cell average is away from the boundary of the convex set. A generic algorithm incorporating the IRP limiter is presented for high order finite volume type schemes. For arbitrarily high order discontinuous Galerkin (DG) schemes to hyperbolic conservation law systems, suf- ficient conditions are obtained for cell averages to remain in the invariant region provided the projected one-dimensional system shares the same invariant region as the full multi-dimensional hyperbolic system does. The general results are then applied to compressible Euler equations so to obtain high order IRP DG schemes. [1] Hailiang Liu and Zhongming Wang, A free energy satisfying discontinuous Galerkin method for one-dimensional Poisson?Nernst?Planck systems, Journal of Computational Physics 328 (2017), 413?437. [2] Yi Jiang and Hailiang Liu, Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations, Journal of Computational Physics (in press) (2018). |
| Roman | Shvydkoy | Unconditional alignment in models of collective dynamics with local interactions | In this talk we present recent developments in regularity and long time behavior of hydrodynamic systems of collective behavior. A common challenge is to construct a model which includes realistic local interactions between agents and at the same time to build an analytically tractable PDE. We present a new class of Cucker-Smale- type systems which utilize only local communication kernels and yet provide a mechanism for unconditional alignment of flocks. The model raises new regularity questions for a class of fractional parabolic equations with non-symmetric rough kernels and drifts. This is a joint work with E. Tadmor. |
| Edriss | Titi | On Recent Advances Concerning the 3D Euler Equations | In this talk I will present some of the recent advances concerning the 3D Euler equations of incompressible flows. |
| Vlad | Vicol | Nonuniqueness of weak solutions to the Navier-Stokes equations | We prove that weak solutions of the Navier-Stokes equations are not unique in the class of weak/mild solutions with finite kinetic energy. This is joint work with Tristan Buckmaster. |
| Ya-Guang | Wang | Stability and separation of the Prandtl boundary layers | In this talk, we study the stability and separation of the Prandtl boundary layers. For the three dimensional Prandtl equations, we deduce that the shear flow is stable if and only if the tangential flow direction is invariant with respect to the normal variable of the boundary, then the well-posedness of the three dimensional Prandtl equations is obtained under a constraint on its flow structure, which avoids the appearance of the secondary flow. Finally, we present a separation result of the unsteady Prandtl boundary layer under adverse pressure gradient. |
Contributed Talks
| First Name | Last Name | Title | Abstract |
|---|---|---|---|
| Debora | Amadori | Decay properties of a nonlinear damped semilinear wave equation on a 1-d bounded domain | We consider a semilinear hyperbolic system with space-dependent and nonlinear damping, on a bounded interval. The damping term is possibly localized in a subinterval. As the solution converges exponentially fast towards a stationary profile, it is interesting to consider approximations that are qualitatively accurate for large times. In this talk we will present a class of approximate solutions (that may also serve as a numerical scheme) and their spectral analysis, which provides information on their asymptotic behavior. We present a new approach that exploits several tools from matrix analysis. Joint work with F. Aqel and E. Dal Santo (University of L'Aquila, Italy) |
| Fabio | Ancona | On Kolmogorov entropy compactness estimates for scalar conservation laws | Inspired by a question posed by Lax [4, 5], in recent years it has received an increasing attention the study of quantitative compactness estimates for the solution operator S_t,t>0 that associates to every given initial data u0 the corresponding solution Stu0 of a conservation law or of a first order Hamilton-Jacobi equation. In particular, in the case of scalar conservation laws u_t+f(u)_x=0, t≥0,x∈R, with uniformly strictly convex flux f, quantitative compactness estimates - in terms of Kolmogorov entropy in L_loc^1 - were established in [2, 3]. We will present an extension of such estimates to the case of scalar conservation laws with a smooth function f that either is strictly (but not necessarily uniformly) convex or has a single inflection point with a polynomial degeneracy. Estimates of this type play a central roles in various areas of information theory and statistics as well as of ergodic and learning theory. In the present setting, this concept could provide a measure of the order of "resolution" of a numerical method for the corresponding equation. |
| Denise | Aregba-Driollet | Dissipation properties of the Chapman-Enskog expansion for a BGK approximation of the bitemperature Euler model | In a recent work ([1]) some relaxation approximations have been developed for a nonconservative Euler system modeling a plasma with a constant ionization. This situation corresponds to the quasi-neutral regime. Here, nonconservativity is due to the presence of products of the velocity by electronic and ionic pressure gradients, and to source terms. In particular, a BGK model coupled with Maxwell-Ampere and Poisson equations through the electric field has been developed and proved to be compatible with the entropy of our system. We study here the Navier-Stokes system obtained by the Chapman-Enskog expansion for this BGK model. For systems of conservation laws, it is a known fact that the second order terms of the Navier-Stokes system coming from the Chapman Enskog expansion of a BGK model are dissipative with respect to the entropy, see [2]. We prove such a result for our system. Work in collaboration with S. Brull (IMB and Bordeaux-INP). [1] D. Aregba-Driollet, J. Breil, S. Brull, B. Dubroca and E. Estibals. Modelling and approximation for the bitemperature Euler model. To appear in M2AN Math. Model. Numer. Anal. [2] F. Bouchut. Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999). |
| Fumioki | ASAKURA | System of Ionized Mixed Gas Dynamics | The aim of this talk is to study a system of three equations of motion for mixed ionized gas, for example Helium and Hydrogen. We suppose that the gas is well mixed so that the mass density and the pressure is a weighted average of each gas. In addition to the mass density, pressure and particle velocity, the degree of ionization of each gas has to be taken into account. New equation of state and the enthalpy function are introduced. By assuming the local thermodynamic equilibrium is attained, the system is closed by requiring the Saha ionization conditions for the both gas. The physical entropy function is constructed. The characteristic speeds are expressed by derivatives of the entropy function. Geometric structure of the system is complicated. We will show the loss of genuine nonlinearity of each characteristic direction. However, Huginiot loci issuing from actual laboratory states are expected to stay within a genuinely nonlinear region. |
| Paola | Bacigaluppi | A Posteriori Limited Robust and Accurate Residual Distribution Schemes for Transient Simulations of Fluid Flows | In the present work, we propose a blended high order Residual Distribution (RD) method in the framework of multidimensional time-dependent hyperbolic systems of equations. The main goal is to maintain high order of accuracy in smooth regions of the flow, while ensuring robustness and a non-oscillatory behavior across strong interacting discontinuities. Following the so-called Multi-dimension Optimal Order Detection (MOOD) approach (Diot et al., Computers and Fluids 2012), a high order accurate candidate solution is replaced by the approximation of a more dissipative residual distribution scheme at a certain time step, if a numerical oscillation is detected. The strengths of the proposed approximation rely on the generic spatial discretization of the model equations using a continuous finite element type approximation technique, while avoiding the solution of a large linear system with a sparse mass matrix which would come along with any standard ODE solver in a classical finite element approach to advance the solution in time. The high order of accuracy both in space and time is achieved via an arbitrary explicit high order approximation where the design of the scheme allows for an efficient diagonalization of the mass matrix without any loss of accuracy. This is achieved by coupling the RD formulation (Deconinck and Ricchiuto, Encyclopedia of Computational Mechanics 2007; Ricchiuto and Abgrall, Journal of Computational Physics 2010; Abgrall. Computers and Fluids 2006) with a Deferred Correction (DeC) type method (Dutt et al., BIT Numerical Mathematics 2000; Liu et al., Journal of Computational Mathematics 2008; Minion, Communication in Mathematical Physics 2003) for the discretization in time and choosing Bernstein polynomials as shape functions to ensure positivity of the resulting diagonal matrix coefficients. In this work, we extend (Abgrall et al., ENUMATH 2015; Abgrall, Journal of Scientific Computing 2017; Abgrall et al., Submitted) to obtain a more robust and accurate scheme and propose a family of blended robust, positivity preserving, high accurate, non-oscillatory and effective residual distribution schemes. This novel scheme has been extensively tested and the considered numerical examples in 1D and 2D assess the effective order of accuracy for smooth flows, the non-oscillatory behavior on flows with strong interacting discontinuities, and both the robustness and positivity preservation on more extreme flows. |
| Wasilij | Barsukow | Stationarity preserving schemes for the linearized Euler equations in multiple spatial dimensions | In multiple spatial dimensions, the Euler equations exhibit a number of phenomena not present in one-dimensional flow, e.g. vortex structures and the incompressible (low Mach number) limit, which only is nontrivial in multiple dimensions. Numerical methods need to be able to resolve these efficiently. The approach of this talk is to first study in detail a simpler system of equations: linear acoustics obtained as linearized Euler equations. It can be endowed with a low Mach number limit and possesses a stationary vorticity. Despite their linearity, these equations pose challenges to standard numerical methods. For instance, the upwind/Roe scheme applied to linear acoustics fails to resolve the incompressible limit, and does not possess any stationary discrete vorticity. This is similar to the behaviour of such methods for the Euler equations. The artefacts as observed for linear acoustics are traced back to the methods not being able to ensure that data corresponding to analytical stationary states remain stationary at discrete level. On the other hand, schemes whose stationary states discretize all the stationary states of the PDE are found to perform much better. They are termed stationarity preserving. This property has a number of implications. It is shown that stationarity preservation is equivalent to the scheme being vorticity preserving, as well as being able to resolve the low Mach number limit. It is shown that truly multi-dimensional stencils appear naturally. The analysis can be used constructively in a variety of situations. |
| Lydia | Bieri | The Einstein Equations and Gravitational Waves | In Mathematical General Relativity (GR) the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear pde has served as a playground for all kinds of new problems and methods in pde analysis and geometry. A major goal in the study of these equations is to investigate the analytic properties and geometries of the solution spacetimes. In particular, fluctuations of the curvature of the spacetime, known as gravitational waves, have been a highly active research topic. In 2015, gravitational waves were observed for the first time by Advanced LIGO (and several times since then). Understanding gravitational radiation is tightly interwoven with the study of the Cauchy problem in GR. I will talk about geometric-analytic results on gravitational radiation and the memory effect of gravitational waves. We will connect the mathematical findings to experiments. I will also address recent work with David Garfinkle on gravitational radiation in asymptotically flat as well as cosmological spacetimes. |
| Paolo | Bonicatto | Untangling of trajectories for non-smooth vector fields and Bressan's Compactness Conjecture | Given $latex d \ge 1$, $latex T>0$ and a vector field $latex \mathbf b \colon [0,T] \times \mathbb R^d \to \mathbb R^d$, we study the problem of uniqueness of weak solutions to the associated transport equation $latex \partial_t u + \mathbf b \cdot \nabla u=0$ where $latex u \colon [0,T] \times \mathbb R^d \to \mathbb R$ is an unknown scalar function. In the classical setting, the method of characteristics is available and provides an explicit formula for the solution of the PDE, in terms of the flow of the vector field $latex \mathbf b$. However, when we drop regularity assumptions on the velocity field, uniqueness is in general lost. In the talk we will present an approach to the problem of uniqueness based on the concept of Lagrangian representation. This tool allows to represent a suitable class of vector fields as superposition of trajectories: we will then give local conditions to ensure that this representation induces a partition of the space-time made up of disjoint trajectories, along which the PDE can be disintegrated into a family of 1-dimensional equations. We will finally show that if $latex \mathbf b$ is locally of class $latex \BV$ in the space variable, the decomposition satisfies this local structural assumption: this yields in particular the renormalization property for nearly incompressible $latex \BV$ vector fields and thus gives a positive answer to the (weak) Bressan's Compactness Conjecture. This is a joint work with S. Bianchini. |
| Stéphane | Brull | ON THE CHAPMAN-ENSKOG ASYMPTOTICS FOR A MIXTURE OF MONOATOMIC AND POLYATOMIC RAREFIED GASES | Collaboration with: C.Baranger, M.Bisi, L.Desvillettes. We propose a derivation of the Chapman-Enskog asymptotics for a mixture of monoatomic and polyatomic gases. We use a direct extension of the model devised for treating the internal energy with only one continuous parameter. This model is based on the Borgnakke-Larsen procedure. We detail the dissipative terms related to the interaction between the gradients of temperature and the gradients of concentrations (Dufour and Soret effects), and present a complete explicit computation in one case when such a computation is possible, that is when all cross sections in the Boltzmann equation are constants. |
| Annegret | Burtscher | On the formation of singularities in Einstein-Euler spacetimes | The general theory of relativity describes the effect of gravitation in terms of the geometry of spacetimes via the Einstein equations. While the local existence of solution is well established, the global structure of solutions is much less understood. In general, singularities are unavoidable. The Penrose singularity theorem relates the presence of black holes to the existence of trapped surfaces. In this talk we will describe how such trapped surfaces can form during evolution, illustrated for spherically symmetric solutions of the Einstein-Euler equations with linear equation of state [1]. The initial data required to observe such trapping phenomena are static solutions with certain admissible large localized perturbations. The local existence result is based on a generalized Glimm scheme. Static solutions themselves arise as solutions to the highly nonlinear, singular Tolman-Oppenheimer-Volkhoff equation. While global existence, uniqueness and even smoothness of these static solutions is well known, very little is still known about their geometry. In general, it is even difficult to determine whether the fluid is confined to a bounded region (with vacuum exterior) or is unbounded. We will see that the asymptotic behavior largely depends on the prescribed equation of state and central density. Solutions with linear and certain polytropic-type equation of state, for example, do not have finite extend and are not even asymptotically flat. We introduce new tools to describe their global geometry [2], which eventually will allow us to identify more general initial data sets that lead to the formation of trapped surfaces. References: [1] A.Y. Burtscher and P.G. LeFloch, The formation of trapped surfaces in spherically-symmetric Einstein-Euler spacetimes with bounded variation, J. Math. Pures Appl. 102 (2014), no. 6, 1164?1217. [2] L. Andersson and A.Y. Burtscher, On the asymptotic behavior of static perfect fluids, Preprint (2018), arXiv:1801.06614. |
| Raimund | Bürger | On the efficient computation of smoothness indicators for optimal-order WENO reconstructions | This contribution is related to a simplified version of weighted essentially non-oscillatory (WENO) schemes for hyperbolic systems of conservation laws. The simplification is based on a set of newly designed smoothness indicators whose computational cost is lower than that of the conventional Jiang-Shu indicators. These smoothness indicators can be used in the context of schemes of the Yamaleev-Carpenter type and the accuracy properties of the resulting non-linear weights are the same as those obtained when using the traditional Jiang-Shu smoothness indicators. The increase of the efficiency and ease of implementation are demonstrated. Moreover, the ease of computation of the new smoothness indicators on unstructured meshes will be briefly addressed. This contribution is based on joint work with Antonio Baeza, Pep Mulet (both Universitat de Val?ncia, Spain) and David Zor?o (Universidad de Concepci?n, Chile). |
| Carey | Caginalp | Shocks and Conservation Laws with Random Initial Conditions | We consider a nonlinear conservation laws with random initial conditions. As shocks are produced even with smooth data, the introduction of random initial conditions such as Brownian motion presents a challenging problem in terms of the density of shocks at arbitrary x and t. I consider a minimization approach (Hopf-Lax) which avoids tracking individual shocks and relates the problem to Hamilton-Jacobi equations. By discretizing the Brownian motion, one can obtain a closed-form expression in terms of n nested Gaussian integrals for the discretized in initial condition problem. One can show convergence in a probabilistic sense to the continuous problem. Some properties of the variance of the solution are also obtained. |
| Laura | Caravenna | Structure and regularity for BV entropy solutions to hyperbolic systems of balance laws | The talk concerns the qualitative structure of BV entropy solutions of a strictly hyperbolic system of balance laws with characteristic fields either piecewise genuinely non-linear or linearly degenerate: we describe the wave-front structure of a BV entropy solution, approximating it by means of a fractional step scheme running with wave-front tracking. As an application, we might discuss the SBV-like regularity of the BV entropy solution. This joint work with Fabio Ancona and Andrea Marson (Padova) extends corresponding results obtained by Bianchini, Bressan, the author, LeFloch and Yu for the case of systems of conservation laws. |
| Julio | Careaga | Solutions by the method of characteristics and flux identification of a non-convex scalar conservation law modeling sedimentation in a cone. | The sedimentation of an ideal suspension in a vessel with variable cross-sectional area can be described by an initial-boundary value problem for a scalar nonlinear hyperbolic conservation law with a non-convex flux function and an area function that depends on spatial position. The sought unknown is the local solids volume fraction with respect to depth and time and it exhibits discontinuities curved in the space-time plane. The behaviors of the entropy solutions depending on different initial concentrations for a parametric cross-sectional area are presented. The inverse problem (IP) is defined as the determination of the flux function of the differential equation given the suspension-liquid discontinuity. It is shown a closed form solution of IP for the parametric cross-sectional area and the main case of a conical vessel. An algorithm necessary for the identification from discrete data, experiments with real data and simulations with the identified flux function using the numerical method described in [1] are presented. This work is based on the joint work [2, 3] with Raimund B?rger (Universidad de Concepci?n, Chile) and Stefan Diehl (Lund University, Sweden). References: [1] R. B?rger, J. Careaga and S. Diehl. A simulation model for settling tanks with varying cross-sectional area. Chem. Eng. Commun. 204, 1270-1281, 2017. [2] R. B?rger, J. Careaga and S. Diehl. Entropy solutions of a scalar conservation law modeling sedimentation in vessels with varying cross-sectional area. SIAM J. Appl. Math. 77: 789-811, 2017. [3] R. B?rger, J. Careaga and S. Diehl. Flux identification for scalar conservation laws modelling sedimentation in vessels with varying cross-sectional area. Preprint 2016-40, Centro de Investigaci?n en Ingenier??a Matem?tica, Universidad de Concepci?n; submitted. |
| Hugo | Carrillo | Lax-Wendroff type reconstruction-free high-order shock-capturing finite difference methods for hyperbolic conservation laws. | We present an order-adaptative finite difference numerical method for systems of conservation laws. The method uses centered 2p+1 stencils, where p may take values in {1,2,…,p_max} according to a family of smooth indicators in the stencils. For linear problems, the method writes as a combination between a robust first order numerical method and the 2p-order generalized Lax-Wendroff methods, so that the numerical method is first order near shocks and of order 2p in smooth regions, where 2p+1 is the size of the biggest stencil in which large gradients are not detected. An algorithm to compute the coeffcients of the method, the choice of the smooth indicators, and the stability analysis (based on [2]) will be presented. For nonlinear problems, the original LW procedure requires the conversion of the time derivatives to spatial derivatives through the so-called Cauchy-Kovalevskaya process, what may increase dramatically the computational cost (see [3]). To avoid this, we adapt the strategy of the Approximate Taylor methods introduced in [4]. The main difference is that, unlike [4], a WENO reconstruction (see [1]) is not needed to ensure the stability of the method, what reduces significantly the computational cost. The general structure of the method and some numerical tests will be shown, in which the results are compared with those provided by standard WENO and Approximate Taylor methods. References: [1] C.-W. Shu. Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws. NASA/CR-97-206253 ICASE Report No. 97-65 (1997). [2] J.Li, Z. Yang. The von Newmann analysis and modi_ed equation approach for finite difference schemes . Appl. Math. Comp. 225 (2013) 610-612. [3] J.Qiu, C.-W. Shu. Finite difference WENO schemes with Lax-Wendroff-Type time discretizations. SIAM J. Sci. Comp. 24 (2003), 2185-2198. [4] D. Zorio, A. Baeza, P. Mulet. An Approximate Lax-Wendroff-Type Procedure for High Order Accurate schemes for Hyperbolic Conservation Laws. J. Sci Comput 71(2017), 246{273. |
| NEELABJA | CHATTERJEE | Convergence analysis of a numerical scheme for a general class of mean-field equations | A widely used prototype phase model to describe the synchronous behavior of weakly coupled limit-cycle oscillators is the Kuramoto model whose dynamics for sufficiently large ensemble of oscillators can be effectively approximated by the corresponding mean-field equation 'the Kuramoto Sakaguchi Equation'. In the recent past, it has been extensively studied to analyze the phase transition of between different kind of ordered states.In the talk, we are going to derive and analyze a numerical method for a general class of mean-field equations, including the Kuramoto Sakaguchi equation. Along the way, we will prove the strong convergence of the scheme to the unique weak solution whenever the initial datum has bounded variation. We also show convergence in the sense of measures, thereby relaxing the assumption of bounded variation. The theoretical results will be verified with several numerical experiments. This is a joint work with U. S. Fjordholm. |
| Geng | Chen | Wave model for nematic liquid crystals | In this talk, we will introduce the recent progress on the global well-posedness on some wave models for nematic liquid crystals. |
| Felisia Angela | Chiarello | General non-local traffic flow models | Conservation laws with non-local flux have drawn growing attention in the recent years because they turned out to be suitable for modeling several phenomena, for example in models for sedimentation, granular flows, supply chains, conveyor belts, population dynamics and traffic flows. In this talk, I will consider the framework of the non-local traffic flow models. I will prove the well-posedness of entropy weak solutions for a class of scalar conservation laws with non-local flux, approximating the problem by Lax-Friedrichs scheme and providing L? and BV estimates for the sequence of approximate solutions, and the stability with respect to the initial data, obtained from the entropy condition through the doubling of variable technique. I will show some numerical simulations about the limit model analitically obtained as the kernel support tends to infinity. Finally, I will prove the stability of entropy weak solutions with respect to the kernel function and the speed, under more regular assumptions and I will provide some numerical simulations to illustrate the dependecies above for some cost functional for traffic flow applications. Joint works with Paola Goatin and Elena Rossi from Inria Sophia Antipolis - M?diterran?e. References: F. A. Chiarello, P. Goatin. Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. 2017. To appear on esaim:m2an. F. A. Chiarello, P. Goatin, E. Rossi. Stability estimates for non-local scalar conservation laws. 2018. [preprint] |
| Maria Teresa | Chiri | On attainable sets for scalar conservation laws with discontinuous flux | Consider the initial value problem for a scalar conservation law (*) u_t + f(x, u)_x = 0; u(x, 0) = u_0, t > 0, x in R; where u = u(t; x) is the state variable and f is a discontinuous flux in the space variable x of the form f(x, u) = f_l(u)1_{x<0}+f_r(u)1_{x>0}, with f_l and f_r strictly convex smooth maps. Equations of this kind typically appear in modelling two phase flows in porous media, in sedimentation problems and in vehicular traffic problems. Here, we are interested in a control theoretic analysis of (*) where we regard the initial data u_0 as a control varying in a given set U of L^{\infty}(R). Namely, we provide a characterization of the sets of profiles attained by: (i) entropy weak solutions of (*) at a fixed time T > 0: A_u(T, U) ={u(-, T) : u is a solution of (*); u_0 \in U}, (ii) flux of entropy weak solutions of (*) at the discontinuity interface x = 0: A_{f_l}(0, U) ={f_l(u(0; -)) : u is a solution of (*), u_0 in U}; A_{f_r}(0, U) ={f_r(u(0; -)) : u is a solution of (*), u_0 in U}; where f_l(u(0, -)), f_r(u(0, -)) denote, respectively, the left and right pointwise traces of the fluxes f_l, f_r at x = 0. The analysis of A_u and A_{f_l},A_{f_r} is carried out relying on the theory of generalized characteristics of Dafermos extended here to the setting of conservation laws with a discontinuous flux. As a consequence, we derive compactness properties in the L^{1}_loc-topology of the above attainable sets and the existence of solutions for a class of optimisation problems in which the cost functional depends on the attainable profiles belonging to the sets A_u, A_{f_l}, A_{f_r}. These results represent also a first step towards the analysis of similar controllability problems for conservation law models of a junction of a network. This is a joint research with Prof. Fabio Ancona (Universit? di Padova) |
| Cleopatra | Christoforou | Stability in the Large of BV Solutions for a Model of Granular Flow | We consider a model of granular flow that consists of a hyperbolic system of two balance laws in one-space dimension. Stability in $L^1$ of entropy weak solutions of bounded variation with respect to the initial data is established for large data up to any finite time $T$. This problem is characterized by the following special features: (i) the characteristic fields are neither genuine nonlinear, nor linear degenerate, (ii) the source terms of the hyperbolic system are non-dissipative and (iii) the initial data are not necessarily of small total variation. Our analysis relies on the extension of the Bressan-Liu-Yang functional to the regime of hyperbolic systems of balance laws under the challenges that are associated with these features and the existence of large BV solutions to this system by Amadori-Shen. Co-Authors: Fabio Ancona and Laura Caravenna |
| Gennaro | Ciampa | Smooth approximation is not a selection principle for transport equation | In our work we are interested in the uniqueness of solutions to the Cauchy problem for the linear transport equation. In the literature there exist various examples of vector fields for which solutions are not unique, when the vector field is less regular then Sobolev or BV. We give a new example of non uniqueness for the transport equation, based on a counterexample of R.J. DiPerna and P.-L. Lions, and we prove that infinitely many solutions can be obtained as limit of solutions of the Cauchy problem with a regularized vector field. Actually there are approximating sequences possessing subsequences converging to different solutions. This is a joint work with Gianluca Crippa and Stefano Spirito. |
| Rinaldo M. | Colombo | Consensus Games via Conservation Laws | We propose a well posed framework to deal with consensus issues through conservation laws. A variety of new control problems naturally arises, requiring the development of new techniques. A straightforward follow up leads to consider several competing controllers, obtaining a game in a framework naturally based on the theory of hyperbolic conservation laws. The present talk presents the analytic framework, numerical integrations, basic well posedness results and open questions. |
| Clémentine | Courtès | A high-order unconditional stable relaxation scheme | KEY WORDS : Lattice-Boltzmann scheme, kinetic BGK model, low-Mach modelization ABSTRACT : The Lattice-Boltzmann methods are used to simulate equations at macroscopic scale from latent microscopic dynamics and are, for example, widely used to model the flows of complex fluids. The discretization of the kinematic models underlying the studied macroscopic problem is conventionally done by means of a propagation-collision scheme. At each time step, two steps are performed: a transport step (or propagation) and a relaxation step (or collision). During the transport step, the particles are restricted to moving from nodes to nodes to the network and discrete space- and time- steps are therefore strongly bounded by a Courant-Friedrichs-Lewy (CFL) condition. In this presentation, we present a kinetic relaxation scheme generalizing these Lattice-Boltzmann methods [1, 2] using a semi-lagrangian transport step. This scheme allows for example to get rid of CFL condition, to design a high-order in time and space method quite easily and to treat non cartesian grid. At the end, we obtain an unconditional stable and high-order scheme without matrices inversion and storage. We will give stability and consistence results on simple cases, show some numerical results on hyperbolic equations or diffusion problems and we will focus on the simulation of low-Mach Euler equations. [1] D. Coulette, E. Franck, P. Helluy, A. Ratnani and E. Sonnendr?cker, Implicit time schemes for compressible fluid models based on relaxation methods, working paper or preprint, hal-01514593, 2017. [2] D. Coulette, E. Franck, P. Helluy, M. Mehrenberger and L. Navoret, Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation, working paper or preprint, hal-01422922, 2016. |
| Gianluca | Crippa | Loss of regularity for linear transport equations | For a linear transport equation $$ \partial_t u + b \cdot \nabla u = 0 $$ with a Lipschitz velocity field $b$, the classical Cauchy-Lipschitz theory ensures propagation in time of the (Lipschitz) regularity of the initial datum. Although for less regular (Sobolev or $BV$, for instance) velocity fields a well-posedness theory for this equation is by now available (based on seminal results by DiPerna-Lions and Ambrosio), it turns out that the issue of the propagation in time of the regularity is much more delicate. In this talk I will report on a joint work with Alberti and Mazzucato, in which Sobolev velocity fields and smooth initial data are constructed, in such a way that any fractional regularity of the solution is instantaneously destroyed. Connections to mixing phenomena in fluids will also be mentioned. |
| Sara | Daneri | Non-uniqueness for the incompressible Euler equations up to Onsager's critical exponent | Our work is related to Onsager's conjecture, proven by Isett and refined by Buckmaster, De Lellis, Szekelyhidi and Vicol, according to which below H\"older regularity 1/3 there exist solutions of the incompressible Euler equations which dissipate the total kinetic energy of the flow. In a series of papers we deal with the associated initial value problem, with regularity below Onsager's critical exponent. Although in a smooth setting dissipation of the total kinetic energy implies uniqueness of solutions with the same initial datum, we are able to show that there exists a dense set of H\"older 1/3-\eps initial data, each admitting infinitely many H\"older 1/3 dissipative solutions of the Euler equations. In particular, an instance of the $h$-principle holds for the incompressible Euler equations below the critical exponent, similarly to what happens for $C^1$ isometric embeddings of Riemannian manifolds in $\R^n$ . Moreover, in order to prove the density in $L^2$ of such wild initial data we introduce a family of stationary solutions of the Euler equations, the so called Mikado flows, which proved to be the crucial ingredient in Isett's proof of the Onsager's conjecture. Some of the results are a joint work with E. Runa and L. Szekelyhidi. |
| Asha | Dond | Edge patch-wise local projection nonconforming FEM for convection-diffusion problem | The standard Galerkin finite element methods fail to provide a stable and non-oscillatory solution for the convection-dominated diffusion problem. We develop edge patch-wise local projection stabilized nonconforming finite element methods for the convection-diffusion problems. It is a composition of the standard Galerkin finite element method, the patch-wise local projection stabilization and weakly imposed Dirichlet boundary conditions on the discrete solution. We study a priori error analysis with respect to a patch-wise local projection norm. The numerical experiments confirm efficiency of the proposed stabilization technique and validate the theoretical convergence rates. |
| Nikodem | Dymski | The Aw-Rascle-Zhang model with constraints | In this talk, we present the Aw-Rascle-Zhang (ARZ) model within the framework of a local point constraint on the flow and moving bottleneck. The main challenge is the presence of shocks violating the classical Kruzkov or Lax entropy admissibility condition. The idea of ARZ with local point constraint on the flow comes from the paper by P.Goatin and M.Garavello. They introduce two different Riemann solvers: the first one is fully conservative, while the second conserves only the mass. In real life applications, this model describes the situation when obstacles such as construction sites, toll gates or traffic lights are present. The approach was extended to moving bottleneck by P.Goatin, C.Chalons ans S.Villa, by adapting the Riemann solvers from the fix constraint approach. The model consists of a strongly coupled PDE-ODE system, where the ODE gives the trajectory of large slow vehicles, e.g. trucks or busses. Remark that the existence of BV solutions for the first Riemann solver and pointwise constraint is done by B.Andreianov, C.Donadello and M.D.Rosini. We prove the existence of BV solutions for non-conservative Riemann solver. The result is achieved by showing the convergence of a sequence of approximate solutions constructed via the wave front tracking algorithm. The same approach can be used to prove the existence of BV solutions for both Riemann solvers. This work is in collaboration with P.Goatin (Inria Sophia Antipolis) and M.D.Rosini(Maria Curie-Sklodowska University). |
| Volker | Elling | Vortex cusps | Vortex cusps are points where two vortex sheets of opposite circulation meet and form a cusp. Special cases are vortex sheets meeting a slip-condition wall tangentially, as observed in numerous types of Mach reflection. We discuss modelling, numerical approximation and construction of such cusps for incompressible Euler. In particular we give formulas for the cusp exponents as functions of the problem parameters. Numerical schemes are proposed and evaluated by their ability to produce cusps with the correct exponents. |
| Shota | Enomoto | Asymptotic behavior of solutions to the compressible Navier-Stokes equation around space-time periodic flow | We consider asymptotic behavior of solutions to the compressible Navier-Stokes equation around space-time periodic states in an infinite layer of $\mathbb{R}^n$ ($n=2,3$) under the action of a space-time periodic external force. If the external force is sufficiently small, then the compressible Navier-Stokes system has a space-time periodic solution. We show that the space-time periodic states is asymptotically stable if the initial perturbation is sufficiently small. Furthermore, it is shown that the asymptotic leading part is given by a product of a solution of the one-dimensional viscous Burgers equation and a space-time periodic function when $n=2$, and by a product of a solution of the two-dimensional heat equation and a space-time periodic function when $n=3$.\\ This talk is based on a joint work with Professor Yoshiyuki Kagei (Kyushu University) and Mr. Mohamad Nor Azlan. |
| Antonio | Esposito | Systems of PDEs with nonlocal Newtonian interactions in 1D | I will discuss a joint work with J.A. Carrillo (Imperial College London), M. Di Francesco (University of L?Aquila), S. Fagioli (University of L?Aquila), and M. Schmidtchen (Imperial College London). We prove global-in-time existence and uniqueness of solutions of a singular nonlocal interaction system of two species in one spatial dimension. While similar results are already known for two or more dimensions using the method of characteristics, this approach breaks down due to the Non-Lipschitz characteristic velocity fields. For initial data including atomic parts we provide a notion of solutions in terms of the corresponding pseudo-inverse functions. The system can then be posed as an $L^2$ gradient flow giving rise to a notion of solutions for the original system. For absolutely continuous initial data we construct solutions using a minimising movement scheme in the set of probability measures. In addition we show that $L^m$ initial data lead to solutions with $L^m$-regularity and its second order moments are preserved at all times. We conclude the paper constructing two different weak solutions emanating from the same initial data. While this seems contradictory at first, it is consistent with the $L^2$-gradient flow theory we develop, because, as soon as the sub-differential can be characterised, there is just one selected solution. |
| Simone | Fagioli | Systems of cross-diffusion with nonlocal interaction. | Macroscopic models for systems involving diffusion, short-range repulsion, and long-range attraction have been studied extensively in the last decades. In this talk I present the analysis to a system for two species interacting with each other according to different inner- and intra-species attractions. Under suitable conditions on this self- and crosswise attraction an interesting effect can be observed, namely phase separation into neighbouring regions, each of which contains only one of the species. We prove that the intersection of the support of the stationary solutions of the continuum model for the two species has zero Lebesgue measure, while the support of the sum of the two densities is a connected interval. Similar behavior can be shown for a system of cross-diffusion with reaction terms. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum. |
| Xiaoting | Fan | Initial layer problem for the Boussinesq system of Rayleigh-Benard convection with infinite Prandtl number limit | The main purpose of this paper is to study the initial layer problem and the infinite Prandtl number limit of Rayleigh-Benard convection with ill prepared initial data. We use the asymptotic expansion methods of singular perturbation theory and the two-time-scale approach to obtain an exact approximating solution and the convergence rates. |
| Raffaele | Folino | Motion of interfaces for hyperbolic variations of the Allen-Cahn equation | Consider the Allen-Cahn equation, that is a reaction-diffusion equation with reaction term f=-F', where F is a double well potential with wells of equal depth, located at ?1. It is well known that, in the one dimensional case, such equation exhibits the phenomenon of metastability as the diffusion coefficient ? goes to 0. Precisely, (unstable) profiles with N transitions from ?1 to +1 evolve very slowly in time: the solution maintains the (unstable) N-transition layer structure for an exponentially long time, i.e. a time of order (exp{c/?}). In the multidimensional case n?2, we have a different behavior of the solutions as ??0: it is well known that, if the initial datum takes the values +1 and ?1 in the regions ?_+ and ?_-, then the 'interface' connecting ?_+ and ?_- moves with normal velocity equal to the sum of its principal curvatures, that is the interface moves by mean curvature flow. In this case, the interface disappears in a time of order (?^{?2}) . The aim of the talk is to discuss these topics in the case of hyperbolic variations of the Allen-Cahn equation. After briefly recalling the results on the metastable behavior of the solutions in the case n = 1 (mainly taken from [1, 2]), the main part of the talk will be devoted to discuss the limiting behavior of the solutions as ? goes to 0 in the case n=2 or n=3. In particular, we focus the attention on radially symmetric solutions, studying in detail the differences with the classic parabolic case, and we prove that, under appropriate assumptions on the initial data u(?,0) and u_t(?,0), the interface moves by mean curvature as ??0 also in the hyperbolic framework (see [3]). Joint work with: Corrado Lattanzio (University of L?Aquila) and Corrado Mascia (Sapienza, University of Rome). References [1] R. Folino. Slow motion for a hyperbolic variation of Allen-Cahn equation in one space dimension. J. Hyperbolic Differ. Equ., 14 (2017), 1?26. [2] R. Folino, C. Lattanzio and C. Mascia. Metastable dynamics for hyperbolic variations of the Allen-Cahn equation. Commun. Math. Sci., 15 (2017), 2055?2085. [3] R. Folino, C. Lattanzio, and C. Mascia. Motion of interfaces for a damped hyperbolic Allen-Cahn equation. Preprint arXiv:1802.05038, (2018). |
| Thierry | Gallay | Stability of vortices in ideal fluids: the legacy of Kelvin and Rayleigh | The mathematical theory of hydrodynamic stability started in the middle of the 19th century with the study of model examples, such as parallel flows, vortex rings, and surfaces of discontinuity. We focus here on the equally interesting case of columnar vortices, which are axisymmetric stationary flows where the velocity field only depends on the distance to the symmetry axis and has no component in the axial direction. The stability of such flows was first investigated by Kelvin in 1880, for some particular velocity profiles, and the problem benefited from important contributions by Rayleigh in 1880 and 1917. Despite further progress in the 20th century, the only rigorous results available so far are necessary conditions for instability under either two-dimensional or axisymmetric perturbations. The purpose of this talk is to present a recent work in collaboration with D. Smets (Paris), where we prove under mild assumptions that columnar vortices are spectrally stable with respect to general three-dimensional perturbations. The proof relies on a homotopy argument, which allows us to restrict the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis in the complex plane. |
| Sondre Tesdal | Galtung | A discrete Lagrangian scheme for the two-component Camassa?Holm system | %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[]{amsart} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsthm} \usepackage{amsrefs} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} % differentiation commands \newcommand{\diff}[2]{\frac{d #2}{d#1}} \newcommand{\pdiff}[2]{\frac{\partial #2}{\partial#1}} \newcommand{\vdiff}[2]{\frac{\delta #2}{\delta#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{A discrete Lagrangian scheme for the two-component Camassa--Holm system} \author{Sondre Tesdal Galtung} \address{NTNU -- Norwegian University of Science and Technology} \email{sondre.galtung@ntnu.no} \maketitle We develop a discrete scheme for solving the initial value problem for the two-component Camassa--Holm system \cite[Eq. (43)]{Olver1996} in Lagrangian coordinates. This system is bi-Hamiltonian and can be restated as a variational problem with an associated Euler--Lagrange equation. The motivation for considering Lagrangian coordinates is that much of the recently developed theory for this system and the original Camassa--Holm equation utilizes this alternate coordinate system, see e.g.\ \cite{Holden2007, Grunert2012}. This means that we go from considering the unknown velocity $u = u(t,x)$ and density $\rho = \rho(t,x)$ in Eulerian variables $t$ and $x$ with initial data $u_0(x)$ and $\rho_0(x)$ to introducing the characteristics $y$ given by the initial value problem $y_{t} = u(t,y), y|_{t=0} = y_0$ as well as the Lagrangian velocity $U := y_t$ expressed in the variables $t$ and $\xi$. Here $y_0$ will be an initial reparameterization of the real line of our choosing, and we note that the strictly positive initial density has asymptotic value $\lim\limits_{x \to \pm\infty}\rho_0(x) = \rho_\infty$. A relatively straightforward finite difference semi-discretization with a forward difference operator $D$ of the kinetic and potential energy functionals for the variational formulation yields \begin{align*} E_{\text{kin}}^{\text{dis}}(t) &= \frac{1}{2}\Delta\xi \sum_{j\in\Z} \left( U_j^2 + \frac{(DU_{j})^2}{(Dy_j)^2}\right)Dy_j, \\ E_{\text{pot}}^{\text{dis}}(t) &= \frac{\varepsilon}{2}\Delta\xi \sum_{j\in\Z} \left(\frac{\rho_{0,j}}{Dy_j} - \rho_{\infty}\right)^2 Dy_j, \end{align*} where $\Delta\xi$ is a constant grid spacing for the real line, and $y_j, U_j$ and $\rho_{0,j}$ are grid cell values for $y, U$ and $\rho_0$. Moreover, the parameter $\varepsilon >0$ controls the strength of the potential energy. In fact, passing $\varepsilon$ to zero decouples the density $\rho$ from the evolution equation for the velocity $u$ and we recover a discretization of the Camassa--Holm equation. From these discretized functionals we can form a discrete Lagrangian $\mathcal{L}^{\text{dis}}$ satisfying the Euler--Lagrange equation \begin{equation*} \pdiff{t}{} \vdiff{U}{\mathcal{L}^{\text{dis}}} = \vdiff{y}{\mathcal{L}^{\text{dis}}} \end{equation*} which yields an evolution equation for $U_j$. Solutions of the above equation will then admit a time-invariant Hamiltonian $\mathcal{H}^{\text{dis}}$. We present numerical results from applying the above scheme to periodic versions of the two-component system and the Camassa--Holm equation, and compare its performance to existing numerical methods, such as a pseudospectral method presented in \cite{Kalisch2005}. Among our results is that the scheme has consistent experimental convergence rates and performs better than the pseudospectral method for non-smooth initial data such as peakons. A proof of convergence for the proposed scheme is in preparation. This is joint work with Xavier Raynaud (NTNU -- Norwegian University of Science and Technology, SINTEF ICT Applied Mathematics) \begin{thebibliography}{9} \bibitem{Olver1996} P.J.\ Olver and P.\ Rosenau. Tri-{H}amiltonian duality between solitons and solitary-wave solutions having compact support. \textit{Phys. Rev. E (3)}, 53(2):1900--1906, 1996. \bibitem{Holden2007} H.\ Holden and X.\ Raynaud. Global conservative solutions of the {C}amassa--{H}olm equation---a {L}agrangian point of view. \textit{Comm. Partial Differential Equations}, 32(10):1511--1549, 2007. \bibitem{Grunert2012} K.\ Grunert, H.\ Holden, and X.\ Raynaud. Global solutions for the two-component {C}amassa--{H}olm system. \textit{Comm. Partial Differential Equations}, 37(12):2245--2271, 2012. \bibitem{Kalisch2005} H.\ Kalisch and J.\ Lenells. Numerical study of traveling-wave solutions for the Camassa--Holm equation. \textit{Chaos Solitons Fractals}, 25(2):287--298, 2005. \end{thebibliography} \end{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| Mauro | Garavello | Traffic control through autonomous vehicles | We introduce a new mathematical model for controlling car traffic through special vehicles, i.e. autonomous and/or connected ones. Car traffic is described by a macroscopic model, while the dynamics of the special vehicles is described by a microscopic model. More precisely, the Lighthill-Whitham-Richards model~\cite{LW, R} \begin{equation} \label{eq:LWR} \pt \rho + \px f(\rho) = 0 \end{equation} describes the evolution of traffic in a single road. Here $\rho = \rho(t,x)$ denotes the density of traffic at time $t$ and at position $x$, and $f:\R \to \R$ is the flux. Assume that a vehicle (or more vehicles), whose position at time $t$ is described by the function $y = y(t)$, aims at controlling the behavior of traffic. The evolution of such a vehicle is described by the ODE \begin{equation} \dot y(t) = u(t) \end{equation} where $u = u(t)$ is a control function, which selects the desired speed. Following the model proposed by Delle Monache and Goatin in~\cite{DMG}, we consider the following control system \begin{equation} \label{eq:system} \left\{ \begin{array}{l@{\qquad}l} \pt \rho + \px f(\rho) = 0 & t > 0, \, x \in \R \setminus \left\{y(t)\right\} \\ \dot y(t) = u(t) & t > 0 \vspace{.2cm}\\ \rho(t, y(t)) \left[v\left(\rho(t, y(t))\right) - \dot y(t)\right] \le F\left(\dot y(t)\right) & t > 0 \vspace{.2cm}\\ \rho(0, x) = \rho_0(x) & x \in \R \\ y(0) = y_0. \end{array} \right. \end{equation} Here, $\rho_0$ and $y_0$ are the initial condition, $v$ is the average speed, while the function $F$ in the third equation gives the reduction of the flux due to the presence of the special vehicle. In the talk, we discuss about the concept of solution for system~(\ref{eq:system}) and we show that, given a control function $u$, a solution exists. The proof is based on the wave-front tracking technique. This is a joint work with P. Goatin, T. Liard, and B. Piccoli. \medskip \begin{thebibliography}{99} \bibitem{DMG} M. L. Delle Monache, P. Goatin. Scalar conservation laws with moving constraints arising in traffic flow modeling: an existence result. \emph{J. Differential Equations}, 257(11):4015?4029, 2014. \bibitem{LW} M. J. Lighthill, G. B. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roads. \emph{Proc. Roy. Soc. London. Ser. A.}, 229:317?345, 1955. \bibitem{R} P. I. Richards. Shock waves on the highway. \emph{Operations Res.}, 4:42?51, 1956. \end{thebibliography} |
| Jan | Giesselmann | Model Adaptive Numerical Schemes in Compressible Fluid Flows | Most physical processes can be described by a variety of mathematical models. Some models are very detailed but hard to solve numerically while others are less detailed, but easy to solve numerically. In numerical simulations it is desirable to use a discretization of detailed models only where this is really required (to conform with some given error tolerance) and simpler models on the rest of the computational domain. This gives rise to the question of how to determine the sub-domain in which the fine model needs to be used. We address this question in a special situation related to compressible fluid flows around obstacles. It is well known that (for fluids which are nearly inviscid) viscosity can be neglected far away from obstacles so that the flow can be described by Euler equations while in boundary layers close to obstacles viscous effects are very important so that Navier-Stokes equations have to be used. In this context the Euler equations constitute the coarse model and the Navier- Stokes equations are viewed as the fine model. We use the relative entropy method to derive bounds for the difference between solutions to the Euler- and the Navier-Stokes equations which are computable from the Euler solution. We will use these bounds as ?error indicators? on which we base our model adaptation procedure. Wherever the bounds exceed a certain value we use (a discretization of) the Navier-Stokes equations while we use the Euler equations everywhere else. To make such a strategy fully practical for Runge-Kutta Discontinuous Galerkin type schemes we need to combine the relative entropy based estimates with reconstructions of the numerical solution. These were recently developed in the context of a posteriori error estimators for systems of hyperbolic conservation laws. This is joint work with Tristan Pryer (University of Reading). |
| Jan | Glaubitz | The principle of discrete least squares in spectral element approximations | Since it was introduced independently by Gauss (1795) and Legendre (1805), the principle of least squares is omnipresent in many fields of numerical mathematics. The same holds for the more general discrete least squares (DLS) approximations. Yet, the principle of discrete least squares is still of limited use in numerical methods for time-dependent (hyperbolic) partial differential equations. Generalising the concepts of interpolation and pseudo L^2-projections, however, a variety of new tools arises. In this talk, we will investigate these tools and their advantages in constructing (novel) stable high-order methods. Following the results from [MPIM preprint series, 2017-63, (2017)] as well as extending them, we are able to prove entropy stability for linear and non-linear test problems using certain nodal collocation-type discontinuous Galerkin methods on any set of collocation points. To carry entropy stability over to non-linear problems, we note the possibility to enforce interpolation or fixed values at points (one of the above mentioned tools). This allows the construction of entropy stable discretisations without introducing complicated (sometimes even unknown) skew-symmetric correction terms. Finally, we note that stability in the sense of positivity (for instance for the pressure of a system) can also be enforced by incorporating linear inequality constraints in the DLS approximation. This talk is a joined work with Thomas Sonar from the Technical University of Braunschweig (Germany) and Philipp ?ffner from the University of Z?rich (Switzerland). |
| Claus | Goetz | On the cell square entropy stability of recontruction-based DG schemes | A modern class of high-order accurate schemes for the numerical solution of hyperbolic conservation laws are so-called PN-PM schemes (Dumbser et al., J. Comput. Phys., 227: 8209?8253, (2008)). In a PN-PM scheme, the solution is represented in a finite element space of piecewise polynomials of degree N ? 0 (hence the PN in the name of the method) and at each time step before the time evolution is carried out, a high order reconstruction of piecewise polynomials of degree M ? N is computed (hence the PM). In this framework, the pure DG method can be viewed as a PN-PN scheme, while the case P0-P M corresponds to the high order finite volume schemes. For N > 0 and M > N a family of hybrid schemes emerges. We study stability properties of PN-PM schemes and show analytically why these methods are, in general, not L2 -diminishing. To this end, we extend the famous cell square entropy stability result of Jiang and Shu (Jiang and Shu, Math. Comp., 62:531?538 (1994)) for DG methods to the PN-PM case and identify which part in the reconstruction step may cause the instability. With this insight we design a flux limiter that enforces a cell square entropy condition for PN-PM schemes in 1D. |
| Xiaoqian | Gong | Analysis of a Nonlinear Hyperbolic Conservatoin Law with Measure Valued Data | Motivated by a classical model for highly re-entrant semiconductor manufacturing systems, we consider an optimal control problem governed by a nonlinear hyperbolic conservation law which is characterized by the nonlocal feature of the velocity and its flux boundary condition. We report progress on the non-existence of L^1 optimal controls for the transition between equilibria. In the setting of positive finite Borel measures, we formulate a new notion of solution in the spirit of Lagrangian solutions and propose a modified notion of weak solution. We prove the existence and uniqueness of such solutions. |
| Ayoub | Gouasmi | Entropy conservative schemes and the receding flow problem | This work delves into the family of entropy conservative (EC) schemes introduced by Tadmor. The discussion is centered around the Euler equations of fluid mechanics and the receding flow problem extensively studied by Liou. This work is motivated by Liou's recent findings that the abnormal spike in temperature observed with finite-volume schemes is linked to a spurious entropy rise, and that it can be prevented in principle by conserving entropy. While a semi-discrete analysis suggests EC schemes are a good fit, a fully discrete analysis based on Tadmor's framework shows the non-negligible impact of time-integration on the solution behavior. A novel EC time-integration scheme is developed to show that enforcing conservation of entropy at the fully discrete level does not necessarily guarantee a well-behaved solution. |
| Tobias | Grafke | Extreme Events and Metastability in Fluids and Waves | Rare but extreme events often have a dramatic influence on the statistics of stochastic systems, but are notoriously hard to handle both analytically and numerically. In particular in fluid dynamics with its overwhelmingly large number of coupled degrees of freedom, the stochastic forcing interacts with nonlocal nonlinear terms to create coherent structures inducing strong non-Gaussianity. I will present how large deviation theory provides a rigorous framework to quantify these effects, allowing to predict the emergence of extreme events, and similarly transition events in metastable fluid systems (as observed in e.g. atmospheric flows), as well as computing their probability and statistics. The method will be applied to predict extreme events in turbulent flows of Burgers equation, Rogue waves for deep water waves and metastable transitions in atmospheric flows. |
| Katrin | Grunert | Lipschitz metrics for nonlinear PDEs | Solutions of both the Hunter-Saxton and the Camassa-Holm equation might enjoy wave breaking in finite time. This means that even classical solutions, in general, do not exist globally, but only locally in time, since their spatial derivative might become unbounded from below pointwise in finite time, while the solution itself remains bounded. Furthermore, energy concentrates on sets of measure zero when wave breaking occurs. Thus the prolongation of solutions beyond wave breaking is non-unique and depends heavily on how the concentrated energy is manipulated. We are going to show how the stability of conservative solutions, i.e., solutions where the energy is not manipulated at breaking time, can be analyzed by constructing a Lipschitz metric, which is based on Wasserstein metrics, in the case of the Hunter-Saxton and the Camassa-Holm equation. This is joint work with J.A. Carrillo and H. Holden. |
| Jean-luc | Guermond | Second-order invariant domain preserving approximation of the Euler equations using convex limiting | A second-order finite-element-based method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain violating, but entropy consistent, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems. |
| Graziano | Guerra | Conservation laws with discontinuous flux: backward Euler approximations and regulated fluxes | Scalar conservation laws $\partial_{t}u+\partial_{x}f\left(t,x,u\right)=0$ with the flux $f$ discontinuous with respect to the time--space variables $(t,x)$ arise in many applications where the conservation laws describe physical models in rough media. For example, traffic flows with rough road conditions, various polymer flooding models in two phase flow in porous media, etc. An extensive body of recent literature has dealt with fluxes which are discontinuous over a finite number of curves in the $(t,x)$ plane. Usually, existence of solutions is found by constructing approximating sequences and uniqueness is obtained through entropy conditions. Different approximations combined with different entropy conditions may lead to different classes of solutions to the same equation. We are interested in solutions obtained using vanishing viscosity approximations i.e.~solutions to $\partial_{t}u+\partial_{x}f\left(t,x,u\right)=\varepsilon \partial_{xx}u$ with $\varepsilon\to 0$. We show that the Crandall Liggett theory of nonlinear semigroup provides a very elegant framework for proving the existence and the uniqueness of the vanishing viscosity limit when the flux $f(t,x,u)=f(x,u)$ has a single discontinuity at the origin $x=0$. Some examples and counterexamples are given that show, among other things, that there exist fluxes $f(x,u)$, with unbounded total variation with respect to the variable $x$, for which a reasonable solution to the conservation law may not even exist. Then we give a definition of regulated functions in two variables which generalizes the definition of functions with bounded total variation. Finally we show the existence and uniqueness of the vanishing viscosity limit for regulated fluxes. This last result is obtained using three main ingredients: comparison results related to viscous Hamilton--Jacobi equations in one space variable, the known result for fluxes with a single discontinuity at the origin and a compensated compactness argument. These results were obtained in collaboration with Alberto Bressan and Wen Shen |
| Soumil | Gurjar | Numerical simulation of a mixed flowing-powder avalanche | Mixed flowing/powder avalanches are characterised by a fast moving core of heavy ice/snow particles and a powder suspension cloud consisting of fine ice dust. These avalanches are especially dangerous because they reach high velocities and long run-out distances, especially in the cold, steep terrains. The area inundated by the avalanche is difficult to predict because the powder cloud can decouple from the avalanche core and move independently, reaching distances well beyond the reach of the dense core. An accurate prediction of avalanche run-out distances, flow velocities and impact pressures in natural 3D terrain is the driving motivation behind the development of improved snow avalanche dynamics models. Accurate modelling of the pressures associated with the spreading of the cloud is often decisive in many practical applications, for both hazard mapping and back calculations of specific events. The focus of this research is on developing numerical avalanche dynamics models in order to understand the flow of a mixed flowing/powder avalanche under complex terrain conditions. In particular, we concentrate on the simulation of the powder suspension cloud within the avalanche. The avalanche will be represented by a two-layer model that divides it into a flowing core and a powder cloud. The snow core is independently simulated by solving the depth-averaged equations governing avalanche flow with accurate second-order numerical solution schemes. The powder cloud will be simulated by solving the incompressible Navier-Stokes equation wherein the injection boundary conditions for the powder cloud will be obtained using the aforementioned simulation for the snow core. |
| Maria | Han Veiga | Structure preserving schemes for the unsteady compressible Euler system with gravity source term and applications in Astrophysics | The early stages of planet formation can be modeled as a quasi-steady state gas disc orbiting around a central star but this setup is generally hard to evolve numerically, in particular when long term evolution is desired, leading often to numerical schemes with narrow applicability. To this end, it is desirable to design a numerical method that solves the unsteady compressible Euler system with gravity source term which can capture steady states (well balanced property) beyond hydrostatic equilibrium and conserve angular momentum while remaining on cartesian coordinates. I would like to present our progress on this topic: a novel high order well balanced discontinuous Galerkin scheme that captures steady states beyond hydrostatic equilibrium based on work by Dedner and Surville, and a well balanced Residual Distribution method which requires less assumptions on the type of steady state, with an additional angular momentum conservation property. Both these schemes solve the multi-dimensional Euler system with a gravity source term. Finally, we show simple analytical solutions which can be useful to guide code development and testing, and show the applicability of the first scheme to simulate protoplanetary discs and planet-disc interactions. |
| Boris | Haspot | Vortex solutions for the compressible Navier-Stokes equations with general viscosity coefficients in 1D: regularizing effects or not on the density | We consider Navier-Stokes equations for compressible viscous fluids in the one-dimensional case with general viscosity coefficients. We prove the existence of global weak solution when the initial momentum $\rho_0 u_0$ belongs to the set of the finite measure ${\cal M}(\R)$ and when the initial density $\rho_0$ is in the set of bounded variation functions $BV(\R)$. In particular it includes at the same time the case of initial momentum which are Dirac masses and initial density which admit shocks. We can observe in particular that this type of initial data have infinite energy. Furthermore we show that if the coupling between the density and the velocity is sufficiently strong then the initial density which admits initially shocks is instantaneously regularizing and becomes continuous. This coupling is expressed via the regularity of the so called effective velocity $v=u+\frac{\mu(\rho)}{\rho^2}\p_x \rho$ with $\mu(\rho)$ the viscosity coefficient. Inversely if the coupling between the initial density and the initial velocity is to weak, then we prove the existence of weak energy in finite time but the density remains a prirori discontinuous on the time interval of existence. |
| Cory | Hauck | Multiscale convergence properties for spectral approximations of a model kinetic equation | We prove rigorous convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by algebraic estimates of the form $N^{-q}$, where $N$ is the number of modes and $q>0$ depends on the regularity of the solution. However, in the multiscale setting, the error estimate can be expressed in terms of the scaling parameter $\epsilon$, which measures the ratio of the mean-free-path to the characteristic domain length. We show that, for isotropic initial conditions, the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More surprisingly, the coefficients of the expansion satisfy super convergence properties. In particular, the error of the $\ell^{th}$ coefficient of the expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and $\mathcal{O}(\epsilon^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. All the above estimates involve constants depending on $N$, the time $t$, and the initial condition. We investigate specifically the dependence on $N$, in order to assess whether increasing $N$ actually yields an additional factor of $\epsilon$ in the error. Numerical tests will also be presented to support the theoretical results. |
| Siming | He | Suppression of blow-up in Patlak-Keller-Segel via shear flows | In this talk, I will present a recent result with Jacob Bedrossian on applying additional shear flow to suppress chemotactic blow-up in parabolic-elliptic Keller-Segel System with super-critical mass in two and three dimensions. |
| Gerardo | Hernandez-Duenas | Two-layer blood flows through axi-symmetric vessels | A model for two-layer blood flows through axi-symmetric vessels will be presented. The model consists of a hyperbolic system of balance laws and its main properties will be discussed. A well-balanced, positivity-preserving central-upwind scheme will be constructed. The merits of the scheme will be tested in a variety of scenarios with applications to problems such as stenoses and aneurysms. |
| Lars Eric | Hientzsch | Low Mach number limit for Quantum Navier-Stokes equations | We consider the low Mach number limit for the system of Quantum Navier-Stokes equations on the whole space in three dimensions for ill-prepared data. Given a sequence of finite energy weak solutions, we show that the density $\rho_{\varepsilon}$ converges to $1$ as the Mach number $\varepsilon$ goes to $0$ while in the limiting system we obtain strong convergence to a Leray solution of the incompressible Navier-Stokes equations. Moreover the gradient part of the velocity field converges strongly to $0$ as $\varepsilon$ goes to $0$. The main novelty is that our analysis only requires the initial data to be of finite energy. We perform a fine analysis of acoustic waves based on some new Strichartz estimates for the linearised system and we exploit \emph{a priori} bounds derived from Bresch-Desjardins type inequalities. This is joint work with P. Antonelli and P. Marcati. |
| Yuchen | Hua | Second order invariant domain preserving central scheme for nonlinear systems of conservation laws | Nonlinear hyperbolic systems of conservation laws have invariant domain properties. Examples of such domains are positivity of density, internal energy, and minimum principle of the specific entropy for the Euler system of gas dynamics. In general, the invariant domain of a hyperbolic system must be a convex set in phase space. Given a second order central scheme, which is possibly invariant domain violating, we modify it via the so-called convex liming to a new invariant domain preserving scheme. The modified scheme preserves the accuracy of the second order method and makes the method robust near vacuum. The limiting framework is valid for arbitrary nonlinear hyperbolic systems in any space dimension. The approach consists of finding a local convex set where the solution will be limited. The construction of the local set is based on a first order Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV), which preserves all the invariant domains of the system. The second order method can be limited to satisfy the convex constraints via slope or flux limiting. The performance for smooth solutions is improved by using a smoothness indicator based on local entropy production. The time stepping is done by using Strong Stability Preserving algorithms and the invariant domain property is proved provided a CFL condition holds. Numerical examples for scalar equations, nonlinear systems in one and two space dimensions will be presented. |
| Ameya | Jagtap | Higher order scheme for Sine-Gordon Equation in Nonlinear Non-homogeneous Media | Sine-Gordon equation is a nonlinear hyperbolic partial differential equation which is the governing equation of wide variety of physical processes, for example, nonlinear dynamics of DNA chain, propagation of magnetic flux in a Josephson junction consisting of two layer of superconducting material separated by an isolating barrier, propagation of dislocation in crystal, fermions modeling in the Thirring model, modulation of weakly unstable baroclinic wave packet in a moving two-layer medium etc. It is well known that the sine-Gordon system describes three different types of rudimentary excitations simultaneously namely, kinks (topological solitons), phonons (continuous waves) and breathers (dynamical solitons) whose dynamics determine the behavior of the system. Solitons are localized waves which retains their shape, energy and velocity while moving. No deformation occurs when two solitons collide with each other but, these colliding solitons suffer a phase shift. In this work higher order scheme is presented for sine-Gordon equation in nonlinear non-homogeneous media. Higher order Legendre spectral element method is used for space discretization which retains all the advantages of spectral and finite element methods. Spectral stability analysis gives implicit expression of critical time step. For both smooth as well as non-smooth density variations, the value of critical time step strongly depends on magnitude of density and its nature of variation. Error analysis gives optimal rate of convergence for the proposed scheme. Various non-homogeneous media are discussed which include nonlinear smooth as well as discontinuous density variations. Time-dependent continuous and discontinuous density variations which can model geodynamical processes like lithospheric deformation, fault dynamics etc are also discussed. Sine-Gordon equation exhibits soliton like solution whose existence and behavior in non-homogeneous media is studied. The spatio-temporal dynamics of these solitons changes in such non-homogeneous medium. Various one and two-dimensional test cases are solved to show the efficacy of the proposed scheme. |
| Katarina | Jegdic | A free boundary problem for the isentropic gas dynamics equations | We consider an initial value problem for the two-dimensional isentropic gas dynamics equations that results in a regular shock reflection with the subsonic state immediate behind the shock. We consider the problem in self-similar coordinates and we obtain a mixed type system for the subsonic flow and the reflected shock. We present existence analysis using the second order elliptic theory and fixed point theory. |
| Ilija | Jegdic | Numerical Simulations Using the Overlapping Grids Method | We consider hyperbolic conservation laws in one dimension and the problem of numerically computing the solution using two overlapping grids. We develop a method, based on the finite element method, and prove that our numerical approximations converge not only to a weak solution, but to the entropy solution. We present numerical results for the Burgers? equation and the Lax shock-tube problem. |
| Jasobanta | Jena | Self-similar Solutions in Gravitational Collapse of Interstellar Gas Clouds | Lie group theoretic method is used to establish the entire class of self-similar solutions to the problem of shock wave propagation through the gravitational collapse of a spherically symmetric interstellar gas cloud. Necessary conditions for the existence of similarity solutions for shocks of arbitrary strength as well as for strong shocks are obtained. The forms of the cooling - heating function for the existence of a self -similar solution has been evaluated. The arbitrary constants occurring in the expressions of the infinitesimals of Lie group of transformations give rise to two different cases of possible solutions i.e. with a power law and exponential shock paths. A particular solution was considered to study the effect of cooling - heating function on the similarity exponent. |
| Yicheng | Jiang | Asymptotic Stability of Traveling Wave for Time-Delayed Reaction-Diffusion Equations | This talk is concerned with time-delayed reaction-diffusion equations on half space $ R_-$ with boundary effect. When the birth rate function is non-monotone, the solution of the delayed equation subjected to appropriate boundary condition is proved to converge time-exponentially to a certain (monotone or non-monotone) traveling wave profile with wave speed $c>c_*$, where $c_*>0$ is the minimum wave speed, when the initial data is a small perturbation around the wave, while the wave is shifted far away from the boundary so that the boundary layer is sufficiently small. The adopted method is the technical weighted-energy method with some new flavors to handle the boundary terms. However, when the birth rate function is monotone under consideration, then, for all traveling waves with $c>c_*$, no matter what size of the boundary layers is, these monotone traveling waves are always globally stable. The proof approach is the monotone technique and squeeze theorem but with some new development. |
| Song | Jiang | Nonlinear stability/instability in the Rayleigh-Taylor problem of stratified compressible MHD fluids | We establish the stability/instability criteria for the stratified compressible magnetic Rayleigh-Taylor (RT) problem in Lagrangian coordinates. More precisely, under the stability condition E< 1, we show the existence of a unique solution with an algebraic decay in time for the (compressible) magnetic RT problem with proper initial data. The stability result shows that a sufficiently large (impressed) vertical magnetic field can inhibit the growth of the RT instability. On the other hand, if E>1, there exists an unstable solution to the magnetic RT problem in the Hadamard sense. This shows that the RT instability still occurs when the strength of the impressed magnetic field is small or the magnetic field is horizontal with a properly large horizontal period cell. Moreover, by analyzing the stability condition in the magnetic RT problem for vertical magnetic fields, we can observe that the compressibility destroys the stabilizing effect of magnetic fields in the vertical direction. Fortunately, the instability in the vertical direction can be inhibited by the stabilizing effect of the pressure, which also plays an important role in the proof of the stability of the magnetic RT problem. In addition, we extend the results for the magnetic RT problem to the compressible viscoelastic RT problem, and find that the stabilizing effect of elasticity is stronger than that of the magnetic fields. |
| LiMin | Jiang | Quasi-neutral limit and the boundary layer problem of the electro-diffusion models arising in electro-hydrodynamics | Quasi-neutral limit and the boundary layer problem of the electro-diffusion model arising in electro hydrodynamics which is the coupled Navier-Stokes and Nernst-Planck-Poisson equations is studied in three space dimensions with the different two mobilities of charges. For the generally smooth doping profile, quasi-neutral limit is proved by using the weighted energy method coupled with multi-scaling approximating expansion techniques. |
| Hrishikesh | Joshi | Dynamic model adaptation of multi-component chemically reacting flows | When numerically modeling a physical phenomenon the governing equations under some conditions can be simplified. The domain can then be decomposed, such that the complex system is employed where necessary and the simple system is employed where the specified conditions are met. Such a dynamic model adaptation based on an a posteriori error estimation is investigated for hyperbolic systems of conservation laws having a stiff source term in the complex system. An interesting application is multi-component chemically reacting flow. The governing equations for the non-equilibrium mixture consists of the conservation of partial mass for each species and the conservation of momentum and energy. When in chemical non-equilibrium, the contribution of mass production in the partial mass balance needs to be accounted for. This contribution can be described by a non-linear stiff source term, based on mass-action kinetics, as a function of the mass fractions of the species, temperature and pressure. When equilibrium is achieved the stiff source terms vanish and provide algebraic equations which reduce the number of variables in the system of partial differential equations. Thus, a smaller, well-behaved system is obtained. Such a system is numerically cheaper to solve than the complex system. The domain decomposition is carried out based on an error indicator employing relative entropy framework. Discontinuous Galerkin in space and an asymptotic preserving discretization in time is utilized. The relative entropy framework requires a certain regularity of (approximate) solutions, hence reconstructions of the numerical solution, as described in [J. Giesselmann and T. Pryer, A posteriori analysis for dynamic model adaptation problems in convection dominated problems, Math. Models Methods Appl.Sci. (M3AS), 2017, 27, 2381 - 2423], are employed. Furthermore, the coupling conditions at the interface between the two models is studied. Results are shown for numerical test cases of chemically reacting flows. Joint work with Jan Giesselmann (RWTH Aachen University) |
| Stephane | Junca | Blow up for some hyperbolic systems related to fractional BV regularity | Global existence for large data is a challenging problem for hyperbolic systems. For a class of conservation laws arising from chemical engineering, the fractional derivatives s = 1/3 or ? in BV for initial-boundary data are critical for the global existence of entropy solutions. The process of the blow is related to the nonlinearity of eigenvalues. Some generalizations for other hyperbolic systems will be discussed. |
| Mostafa | Kadiri | Optimization of Fish Passage Structures by a Multilayer Model. | In this work we are interested in the design of a vertical slot fishway. This design reasonably handles the seasonal water surface elevation changes. The shape optimization problem is adressed using a multilayer Saint-Venant model as a state system, and the objective function is defined in order to allow fish crossing the obstacle in a convenient conditions. The multilayer Saint-Venant system is a set of coupled modified shallow water systems which allows to consider nonconstant vertical velocities and obtain an accurate description of the vertical profile of the horizontal velocity. A Roe finite volume method combined to a derivative-free algorithm are used to approximate the mathematical state system and the related optimization problem. Numerical simulations are given to show the robustness and the computational ef- ficiency of the technique. Keywords. Shape optimization, multilayer system, Roe finite volume method. References [1] Audusse, E. A multilayer Saint-Venant system: Derivation and numerical validation. Discrete Contin. dyn. Syst. Ser. B 5 (2005). [2] Fern´andez-Nieto E.D. , Kon´e E. H., Chac´on T. A multilayer method for the hydrostatic Navier-Stokes equations: a particular weak solution. Journal Scientific Computing, 60 (2014). [3] Kadiri M., Louaked M., Mechkour H. Optimal design of vertical slots fish ladder. Submitted to Mechanics&Industry, (2017). [4] Louaked M., Mechkour H., Kadiri M. Numerical analysis and optimal management of a fishway model. In MOCASIM 2017 Marrakesh, Morocco, (2017). |
| Yoshiyuki | Kagei | Bifurcation of the compressible Taylor vortex | The Couette-Taylor problem, a flow between two concentric rotating cylinders, has been widely studied as a good subject of the study of pattern formation and transition to turbulence. Consider the case where the inner cylinder is rotating with uniform speed and the outer one is at rest. If the rotating speed is sufficiently small, a laminar flow (Couette flow) is stable. When the rotating speed increases, beyond a certain value of the rotating speed, a vortex flow pattern (Taylor vortex) appears. For viscous incompressible fluids, the occurrence of the Taylor vortex was shown to solve a bifurcation problem for the incompressible Navier-Stokes equations. In this talk, this problem will be considered for viscous compressible fluids. The spectrum of the linearized operator around the Couette flow is investigated and the bifurcation of the compressible Taylor vortex is proved when the Mach number is sufficiently small. It is also proved that the compressible Taylor vortex converges to the incompressible one when the Mach number tends to zero. This talk is based on a joint work with Prof. Takaaki Nishida (Kyoto University) and Ms. Yuka Teramoto (Kyushu University). |
| Alexander | Keimer | Nonlocal conservation laws: Existence, Uniqueness and Regularity | In our talk we consider scalar and multi-dimensional nonlocal conservation laws and present some recent results in existence and uniqueness. The term nonlocal refers to the dependency of the flux function on the averaged solution. The averaging, i.e.\ the spatial integration of the solution, can be carried out over the entire domain or might also be spatially dependent. We show that there exists a unique weak solution without prescribing an Entropy condition. In addition, we discuss the possible breakdown of solutions in finite time as well as the regularity of solutions dependent on initial datum (source terms, weights, etc....). From the point of view of applications, nonlocal conservation laws offer a variety of different applications, for instance in the simulation of pedestrian flow, crowd dynamics, traffic flow, supply chains, chemical ripening, and more. |
| Laura | Keller | Homogenization with two kinds of microstructures: From the microscopic to the macroscopic description of concentrations of chemical agents | We investigate how geometric microstructures of a domain can affect the concentration and distribution of e.g. a chemical agent in a cell organelle on the macroscopic level. More precisely, we look at a domain with additional microstructures of two kinds, the first one are periodically arranged 'horizontal barriers' and the second one are 'vertical barriers' which are not periodically arranged, but uniform on certain intervals. Both structures are parametrized in size by a small parameter $\varepsilon$. |
| David | Kerkmann | ADER interpretation of the Active Flux Method | We study the active flux method of Eymann and Roe and present several new results for one-dimensional hyperbolic problems. For linear problems we show that the active flux method is equivalent to a formulation of an ADER method. This interpretation motivates the construction of new third order accurate methods for nonlinear hyperbolic conservation laws. While commonly used methods for hyperbolic problems are based on discontinuous reconstructions, the active flux method uses a continuous, piecewise quadratic reconstruction. For nonlinear problems we identify a situation in which the continuous reconstruction leads to an unstable approximation. We propose a limiting strategy which overcomes this problem. Our limited version of the active flux method uses the same local stencil as the original method. We also discuss the use of the Active Flux Method in a large time step method. |
| Jeongho | Kim | Global Cauchy problem for a coupled system of kinetic TCS model with Navier-Stokes flow | We study the global-in-time existence theory for the kinetic thermomechanical Cucker- Smale equation interacting with incompressible/compressible flows through a drag force in three dimensions. Our main system consists of Vlasov-type equation describing many thermomechanical Cucker-Smale flocking particles and Navier-Stokes equations. We show the global existence of weak solutions and then establish the global existence and uniqueness of strong solutions. We also provide a priori estimate of large-time behavior of solutions showing the exponential alignment between particles and fluid velocities as time evolves. |
| Stephan | Knapp | Production network models with stochastic capacities: semi-Markov and load-dependent approaches | We focus on production network models based on coupled ordinary and partial differential equations combined with time-dependent random capacity functions. The partial differential equations are scalar conservation laws and of hyperbolic type coupled with ordinary differential equations at the boundaries. In a first step, the random capacity function of every processor is an external given stochastic process, a semi-Markov process that allows intermediate capacity states in the range of total breakdown to full capacity. The operating and down times can be arbitrarily distributed, provided they keep positive. The mathematical challenge is to combine the theory of semi-Markov processes within the framework of conservation laws and the solution concept. In general, the assumption that capacity drop probabilities are independent of the production and external given by a stochastic process is too restrictive. A high workload implies a high abrasion or stressed working conditions whereas an empty production is not affected by capacity drops for example. This motivates to introduce an influence from the production to the capacity process as well and we obtain a bidirectional relation between the production and the random capacity process. For this purpose, we define the stochastic load-dependent production network model as a stochastic process in a whole and embed it into the theory of piecewise deterministic Markov processes. We present solution concepts for both stochastic production network models and show the wellposedness. Caused by the complexity of the model, we state suitable simulation methods and performance measures to evaluate and interpret the results. A variety of numerical examples emphasizes the characteristics of the proposed approach. |
| Arun | Koottungal Revi | STIFFLY ACCURATE DISCRETISATIONS OF THE EULER-POISSON SYSTEM IN THE QUASINEUTRAL LIMIT | We consider the Euler-Poisson system in the quasineutral regime. The quasineutral limit can be characterised as a singular limit of the Euler-Poisson system, where a scaled Debye length ? ? 0. It is well-known that in this limit, the compressible Euler-Poisson equations converge to the incompressible Euler equations; see e.g. [1]. The numerical approximation of singular limits of hyperbolic systems faces several challenges. First, the time-steps are severely limited due to stiffness arising from the stability requirements [6]. Second, the presence of fast waves, e.g. the pressure waves, in the solution can give rise to small time-scales where a numerical solution can deviate significantly from a solution of the limit equation, [4, 5]. A sufficient condition to ensure the accuracy of a numerical solution as ? ? 0 is that the solution of the compressible Euler- Poisson system at any time should be close to that of the incompressible system, usually known as the equilibrium solution. We propose high order asymptotic-preserving (AP) IMEX Runge-Kutta schemes for the Euler-Poisson system and analyse their accuracy as ? ? 0; see also [2, 3] and the reference therein. We show that when the initial data are well-prepared, the solution of the AP schemes remain close to the equilibrium solution. A stability analysis confirms that the AP schemes are also asymptotically stable. |
| Thomas | Kugler | Model reduction for wave propagation with nonlinear damping on networks | We consider the structure preserving model reduction of first order hyperbolic systems describing the damped propagation of waves on 1-d networks. The applications we have in mind cover electric transmission lines and the propagation of pressure waves in gas pipelines. These problems share many interesting physical properties, like conservation of charge or mass, passivity, a port-Hamiltonian structure, unique steady states, and exponential stability. In a previous work we considered the systematic dimension reduction by Galerkin projections onto low dimensional subspaces for a linear wave propagation model. We now investigate the systematic extension of our proposed strategy to a system with nonlinear damping. The resulting pair of approximation spaces can be shown to yield a reduced order model with all the desired properties. Our construction is formulated in a functional space setting which additionally leads to results that are independent of the truth approximation for the underlying pde system. Both properties are illustrated by numerical tests. |
| Rakesh | Kumar | Hybrid FDM-WENO Methods for the Convection Diffusion Problems | In the present work, we have proposed high order Hybrid FDM-WENO methods (upto $8^{\mbox{th}}$ order of accuracy) for the solution of convection-diffusion problems. In hybrid FDM-WENO methods, a high order finite difference central flux is used to compute the convective flux in smooth regions whereas in a region where solution has sharp variations or discontinuities, Weighted Essentially Non-Oscillatory (WENO) reconstruction of Adaptive Order is used to maintain a non-oscillatory profile. For the diffusion part, we have used a high order finite difference approximation which is compatible with convection approximation. A weak local residual based estimate is used to detect the discontinuities or high gradient regions of the solution. The accuracy and resolution of the proposed hybrid algorithms are compared with pure WENO schemes of adaptive order for 1D and 2D test problems. The proposed hybrid algorithms are found to be more accurate in $L^{\infty}$- and $L^1$-error norms and computationally efficient than WENO schemes of adaptive order. |
| Alexander | Kurganov | Path-Conservative Central-Upwind Schemes for Nonconservative Hyperbolic Systems | I will present a path-conservative central-upwind scheme for nonconservative hyperbolic systems of PDEs. Such systems arise in a variety of applications and the most challenging part of their numerical discretization is a robust treatment of nonconservative product terms. Godunov-type central-upwind schemes were developed as an efficient, highly accurate and robust ?black-box? solver for hyperbolic systems of conservation laws. They were successfully applied to a large number of hyperbolic systems including such nonconservative systems as two-layer shallow water equations, compressible two-phase flow models, Savage-Hutter type system modelling submarine underwater slides. To overcome the difficulties related to the presence of nonconservative product terms, several special techniques were proposed. However, none of these techniques was sufficiently robust and thus the applicability of the original central-upwind scheme was quite limited. We have recently realized that the main drawback of the original approach was the fact that the jump of the nonconservative product terms across cell interfaces has never been taken into account. Rewriting central-upwind schemes in the form of path-conservative schemes has helped to understand how the nonconservative products should be discretized so that their influence on the numerical solution is accurately taken into account. The resulting path-conservative central-upwind scheme is a new robust tool for both conservative and nonconservative hyperbolic systems. The new scheme has been applied to the Saint-Venant system with discontinuous bottom topography, two-layer shallow water system, and the two-mode shallow water equations which was derived as a simplified model that describes nonlinear dynamics of waves with different vertical profiles. Our numerical results illustrate a superb performance of the new path-conservative central-upwind scheme, its robustness and ability to achieve very high resolution. |
| Marc | Laforest | ||
| Nicolas | LAURENT-BROUTY | The zero relaxation limit for the Aw-Rascle-Zhang traffic flow model | The Aw-Rascle-Zhang model can now be considered as a classical traffic flow model. In this talk we will detail the mathematical behavior of the Aw-Rascle-Zhang model with relaxation. In a Lagrangian setting, we use the Wave-Front-Tracking method with splitting technique to construct a sequence of approximate solutions. We prove that this sequence admits a limit. We then show that the limit is a weak entropy solution of the relaxed system associated to a given initial datum with bounded variation. Finally, we prove that this limit converges to a weak solution of the scalar conservation law when the relaxation parameter goes to zero. Joint work with Paola Goatin (Inria Sophia Antipolis, France). |
| Omar | Lazar | Global well-posdeness for the Muskat Problem in the critical Sobolev space | The Muskat problem is a nonlinear and nonlocal equation that modelises the dynamic of two fluids of different characteristics. In this talk, I shall present a global existence result for the Muskat problem with data in the critical Sobolev space. The proof is based on a new formulation allowing to take advantage of the hidden oscillations and to use Besov type estimates. Joint with D. Cordoba. |
| Jae Min | Lee | Geometric approach on the global conservative solutions of the Camassa-Holm equation | We construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We ?rst express the equation in Lagrangian ?ow variable $\eta$ and then transform it using a simple change of variable $\rho=\sqrt{\eta_x}$. The new variable removes the singularity of the Camassa-Holm equation, and we obtain both global weak conservative solutions and global spatial smoothness of the Lagrangian trajectories, which were originally discovered by Bressan-Constantin and McKean, respectively. This work is inspired by J. Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation. |
| Philippe G. | LeFloch | Recent advances on the nonlinear stability of self-gravitating matter | I will present mathematical results on the nonlinear dynamical stability of matter fields, including the problems of the global evolution of spherically symmetric flows in Einstein and f(R)?gravity, of flow evolution outside of a Schwarzschild black hole, and the global nonlinear stability of Minkowski spacetime for the Einstein equations in presence of massive fields. |
| Marta | Lewicka | ||
| MINGJIE | LI | Low Mach Number Limit of Steady Euler Flows in the Multi-Dimensional Nozzle | In this talk, we justify the low Mach number limit of the steady irrotational Euler flow in the multi-dimensional nozzle. The existence and uniqueness of the incompressible flow is established. Then, the compressible flow converges to the corresponding incompressible one as the Mach number, which is denoted by $\varepsilon$, goes to zero. The limit is on the H\'{o}lder space and is unique. Moreover, the convergence rate is of order $\varepsilon^2$. |
| Haitong | Li | Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping | In this paper, we study the one-dimensional bipolar Euler-Poisson equations with time-dependent damping $-\frac{J}{(1+t)^\lambda}$ for $-1<\lambda<1$, where the damping effect is time-gradually-degenerate for $\lambda>0$, and time-gradually-enhancing for $\lambda<0$. Such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time-asymptotic-weakly or strongly. Based on some technical observation, and by using the time-weighted energy method, where the weights are artfully chosen, we prove that the system admits a unique global smooth solution, which time-asymptotically converges to the corresponding diffusion wave, when the initial perturbation around the diffusion wave is small enough. The convergence rates are specified in the algebraic forms $O(t^{-\frac{3}{4}(1+\lambda)})$ and $O(t^{-(1- \lambda)})$ according to different values of $\lambda$ in $(- 1,\frac{1}{7})$ and $(\frac{1}{7},1)$, respectively, where $\lambda=\frac{1}{7}$ is the critical point, and the convergence rate at the critical point is $O(t^{-\frac{6}{7}}\ln t)$. All these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are $L^2$-integrable. Particularly, when $\lambda=\frac{1}{7}$, the convergence rate is the fastest, namely, the asymptotic profile of the original system at the critical point is the best. |
| Maria | Lukacova | The role of measure-valued solutions in compressible flows. | In this talk we discuss the convergence of suitable numerical schemes for both viscous and inviscid compressible flows. In fluid dynamics existence proofs are typically based on the construction of sequences of approximate solutions. However, if the underlying model does not provide enough information for the required regularity of the approximate sequence, we are facing the problem to show the scheme's convergence. In particular, for multidimensional problems fine scale oscillations persist, which prevents us to obtain compactness result. Consequently, the standard framework of integrable functions seems not to be appropriate in general. To overcome this problem we introduce the class of dissipative measure-valued solutions, which allows us to show the convergence of finite volume or combined finite volume-finite element schemes for multidimensional isentropic Euler and Navier-Stokes equations, respectively. On the other hand, using the weak-strong uniqueness result for the above systems we know, that the dissipative measure-valued solution coincides with the strong solution, if the latter exists. Consequently, our results show convergence of our numerical schemes to the strong solutions. These results have been obtained in collaboration with Eduard Feireisl (Academy of Sciences, Prague) and supported by the Collaborative Research Center TRR 146 in Mainz. |
| Kjetil Olsen | Lye | Computing statistical solutions of hyperbolic conservation laws | An open question in the field of hyperbolic conservation laws is the question of well-posedness. Recent theoretical and numerical evidence have indicated that multidimensional systems of hyperbolic conservation laws exhibit random behavior, even with deterministic initial data. We use the framework of statistical solutions to model this inherit randomness. We review the theory of statistical solutions for conservation laws. Afterwards, we introduce a convergent numerical method for computing the statistical solution of conservation laws. We show convergence in the narrow topology on the space of probability measures. For the scalar case, we validate our theory by computing the structure functions of the Burgers? equation with random initial data. We especially focus on Brownian initial data, and the measurement of the scalings of the structure functions. The results agree well with the theory, and we get the expected convergence rate. We furthermore show that we can get faster computations using Multilevel Monte Carlo for computing the statistical solutions of scalar conservation laws. In the case of systems of equations, we test our theory against the compressible Euler equations in two space dimensions. We check our numerical algorithm against ill-behaved initial data, the Kelvin-Helmholtz instability, the Richtmeyer-Meshkov instability and a flow where the initial data has a Brownian motion type velocity field. For all experiments we observe convergence of the structure functions, as well as Wasserstein convergence for the second correlation marginals. At the end, we evaluate techniques for lowering the computational costs, including applicability of Multilevel Monte Carlo and Quasi Monte Carlo. |
| Jim | Magiera | A Particle-based Riemann Solver for Compressible Liquid-Vapor Flow with Sharp Interfaces | For compressible liquid-vapor flow with a sharp interface between the bulk phases it is important to account for the microscale properties of the fluid at the phase boundary. They can have a major influence on the macroscopic dynamics. On the continuum scale special Riemann problems \cite{menikoff.plohr:riemann:1989} at the interface can be typically solved by four waves. One of these waves, the phase boundary, is an undercompressive shock wave, which is determined by a kinetic relation. However, particularly in case of fluid mixtures, the physically relevant choice for the kinetic relation is often unclear. \newline Taking this as our motivation, we develop a particle-based multiscale solver for Riemann problems that includes microscale effects by applying methods as e.g. in \cite{fechter.munz.ea:sharp:2017,magiera.rohde:particle:2017}. Furthermore, as an ab initio approach this approach does not require to formulate a kinetic relation. To begin with, we consider the Euler equations on the continuum mechanical scale \begin{align*} \partial_t \rho + \nabla \cdot (\rho \vec{v}) & = 0, \\ \partial_t (\rho \vec{v}) + \nabla \cdot \left((\rho \vec{v}) \otimes \vec{v} + p \boldsymbol{I}\right) & = 0, \\ \partial_t E + \nabla \cdot \left( (E + p) \vec{v} \right) &= 0, \end{align*} where $\rho = \rho(x,t)$ denotes the fluid density, $v = v(x,t)$ the fluid velocity, and $E = E(x,t)$ denotes the energy density, satisfying $E = \rho \varepsilon + \tfrac{1}{2} \rho \vec{v} \cdot \vec{v}$. The pressure $p = p(\rho, T)$ and the specific internal energy $\varepsilon = \varepsilon(\rho, T)$ are given functions of the density $\rho$ and temperature $T$, specified by equations of state. In this case we consider the equation of state for the Lennard--Jones fluid \cite{eos:vrabec:16}. \newline Subsequently, we augment the macroscopic model at the phase boundary by data obtained from molecular dynamics simulations. For this purpose, we consider a generic Lennard--Jones fluid, that resembles noble gases, e.g. argon or xenon. \newline To resolve a macroscopic discontinuity, the macroscopic primary variables are translated to their microscopic counterparts, the microscopic particle distributions, using the Irving--Kirkwood formulas \cite{irving.kirkwood:statistical:1950}. Consequently, we obtain Riemann problems on the microscale with molecular density distributions $\rho_{l}$, $\rho_r$, velocity distributions $v_l$, $v_r$, and temperatures $T_l$, $T_r$. For each of these Riemann problems with the initial data as input data, we can set up a molecular dynamics simulation that results in a microscale wave pattern from which we extract the adjacent states of the phase boundary. Those states form the output data of the microscale simulation, and they can be used to compute the flux on the continuum mechanical scale. \\ Despite the fast and efficient computation of the molecular dynamics simulation on GPUs, they require a high computational effort. Therefore, kernel-based model reduction techniques are applied, to decrease the computational time severely. This is possible due to the fact that only the input-output data relation of the microscale model is of interest. In this contribution the aforementioned particle-based Riemann solver for liquid-vapor flow and numerical simulation results of the complete multiscale model are presented. |
| Mailasu | Mailasu | Non-relativistic limit analysis of the Chandrasekhar-Thorne relativistic Euler equations with physical vacuum | In this talk, we are concerned with a first step to make mathematical analysis for the relativistic Euler equations in terms of $\frac{1}{c^2}$ proposed by S. Chandrasekhar and K.S. Thorne. We consider the non-relativistic limit for the local smooth solutions to the free boundary value problem of the cylindrically symmetric relativistic Euler equations, when the mass energy density includes the vacuum states at the free boundary. For large enough (rescaled) speed of light and suitably small time, we obtain uniform, with respect to the speed of light $c$, ?a priori? estimates for the local smooth solutions. Moreover, the smooth solutions of the cylindrically symmetric relativistic Euler equations converge to the solutions of the classical compressible Euler equation, at the rate of order $\frac{1}{c^2}$. This is the joint work with Hailiang Li and Pierangelo Marcati. |
| Tetu | Makino | ||
| Yogiraj | Mantri | A well-balanced scheme for network of gas pipelines with wall friction | Gas flow through pipeline network can be described using 2x2 hyperbolic balance law along with coupling conditions at the nodes of the network. The numerical solution at steady state is highly sensitive to the coupling conditions at the nodes and the balance between the flux and source terms within the pipes. To avoid spurious oscillations for near equilibrium flows, it is essential to design well-balanced schemes. Recently Chertock, Herty & ?zcan(2017) introduced a well-balanced method for 2x2 systems of balance laws. In the present talk, we expand this approach to include coupling conditions and compressors for gas pipeline network. We rigorously prove the well-balancing for single pipe-to-pipe connection and for compressors. Fist numerical experiments demonstrate stability for T-junctions as well. In ongoing work, we are expanding the scheme to network of water channels with variable width and bottom topography. A future goal is to design higher order methods which could resolve real networks with only a few grid-points per pipe. This is joint work with Michael Herty and Sebastian Noelle. |
| Francesca | Marcellini | A Two-Phase Traffic Model in the Description of Traffic | Within the analytic environment provided by a two-phase traffic model, we present descriptions of various specic situations, from a single road segment to a general junction with n incoming and m outgoing roads. A justication of this setting is provided through the homogenization of a model for junctions regulated by traffic lights. In particular, we propose a Riemann solver which conserves, through the junction, also the maximal speed of each vehicle, which is a key feature of the present model. We also consider the presence of moving constraints. In each case, a well posedness result is preliminary to the verification that the analytic properties, specic to the different settings, qualitatively agree with real traffic behaviour. Part of the present results were obtained in collaboration with Mauro Garavello, Thibault Liard and Benedetto Piccoli. |
| Elio | Marconi | Structure and regularity of solutions to 1d scalar conservation laws | We consider bounded entropy solutions to the scalar conservation law in one space dimension: \begin{equation*} u_t+f(u)_x=0. \end{equation*} We introduce the notion of Lagrangian representation for a solution $u$ and we discuss the structure of its characteristics. Moreover we quantify the regularizing effect of the non linearity of the flux $f$ on the solution $u$ in terms of spaces of functions with bounded generalized variation. This is a joint work with Stefano Bianchini. |
| Christoph | Matern | The Riemann problem for a weakly hyperbolic two-phase ow model of dispersed vapor bubbles in a liquid. | We consider Riemann problems for a two phase flow model of a dispersed phase in a compressible carrier phase of the same material. It is a weakly hyperbolic system of partial differential equations in a fully conservative form. This model is the conservation part of a more complete physical model involving phase transitions. The purpose of this talk is to better understand the mathematical properties of the simplified model. We present the characteristic structure of the Riemann problems and the construction of an exact solution. We establish that solutions depend on the initial left and right states for the dispersed phase velocity. They may contain delta shocks or vaporless states. We give examples for initial data corresponding to a system of water bubbles dispersed in liquid water, as well as water droplets in liquid vapor. |
| Fabian | Meyer | A posteriori error analysis for random conservation laws | To account for uncertainties in numerical simulations and keeping track of the resulting outcomes has become an important issue in recent years. A significant aspect when considering uncertainty as an additional parameter is to quantify the errors that arise from numerical discretization. Additionally, it is desirable to distinguish between errors from discretizing the random space and deterministic errors that arise from spatio-temporal discretization. To address this issue we present an a posteriori error analysis for random hyperbolic conservation laws with uncertain initial condition, uncertain model parameters and random source term, where the uncertain input is modeled as random variable $\xi:\Omega \to \mathbb{R}$. The equation of interest then reads as follows: \begin{align*} \partial_t \bm{U}(t,x,\xi(\omega))+ \nabla_x \bm{F}(\bm{U}(t,x,\xi(\omega)),\xi(\omega))=\bm{S}(t,x,\xi(\omega)), \quad &(t,x,\omega) \in \mathbb{R}_{>0}\times [0,1]_{\mathrm{per}}\times\Omega \\ \bm{U}(0,x,\xi(\omega))= \bm{U}^0(x,\xi(\omega)),\quad ~ &x \in [0,1]_{\mathrm{per}}\times\Omega. \end{align*} Besides intrusive methods like the Stochastic Galerkin method \cite{GMR018}, there exist non-intrusive methods to discretize the random space. In this talk we focus on the Stochastic Collocation method and the non-intrusive Polynomial Chaos Method. The physical space is discretized by a Discontinuous Galerkin scheme. The relative entropy stability framework of Dafermos \cite{dafermos2005} combined with the reconstruction of the numerical solution allows us to derive a computable a posteriori error estimator for the random conservation law. It turns out that the corresponding residual admits a decomposition into a spatial and a stochastic residual, which enables us to control the errors arising from spatial and stochastic discretization. Moreover, this splitting gives rise to new adaptive numerical schemes where the spatial and stochastic residual are used as indicators for the local refinement. Besides the theoretical findings we present numerical computations for the Euler equations. |
| Arthur | Miranda | Conservative Lagrangian-Eulerian finite volume formulations for approximating hyperbolic models with source terms | In the current work we present a simple numerical method based on a Lagrangian-Eulerian framework for approximate solutions of nonlinear balance law problems. This framework has been used for dealing with the dynamic forward tracking of the delicate well-balancing between the first-order hyperbolic flux and the source term. The mass conservation takes place in a local space-time control volume, and this region is used a novel approach to define naturally a balance law. This balance law is the central idea to build a efficient numerical method to approximate solution to balance law problems. We do not use approximate or exact Riemann solvers nor nonlinear reconstructions and we do not use upwind source term discretizations either. We present numerical solutions to nontrivial partial differential equations of several types, both hyperbolic conservation laws and balance laws. Verification of the technique is also made by comparison with analytical solutions when they are available. We present our latest results in applying the weak asymptotic solutions technique to this framework. |
| Stefano | Modena | Non-uniqueness for the transport equation with Sobolev vector fields | We consider the linear transport equation $\partial_t \rho + u \cdot \nabla \rho = 0$, with unknown density $\rho$ and given divergence-free vector field $u$, together with a given initial datum $\rho(x,0) = \bar \rho(x)$. A celebrated result by DiPerna and Lions (1989) shows that if \begin{equation*} \bar \rho \in L^p_x, \quad u \in L^1_t W^{1,\tilde p}_x \end{equation*} and \begin{equation*} \frac{1}{p} + \frac{1}{\tilde p} \leq 1, \end{equation*} then the Cauchy problem admits a unique weak solution $\rho \in L^\infty_t L^p_x$. We show that the above condition is optimal: if \begin{equation*} \frac{1}{p} + \frac{1}{\tilde p} > 1, \end{equation*} then there are divergence-free vector fields $u \in C_t W^{1, \tilde p}$ and initial data $\bar \rho \in L^p_x$ for which, despite the linearity of the equation, more than one weak solution in $C_t L^p_x$ exists. The result applies also to the transport-diffusion equation $\partial_t \rho + u \cdot \nabla \rho = \Delta \rho$ (joint with L. Sz\'ekelyhidi). |
| Tomas | Morales de Luna | A fully well balanced Lagrange-Projection splitting scheme for shallow water | Has already been sent by e-mail |
| Alessandro | Morando | Local existence of 2D nonisentropic compressible vortex sheets | We show the local-in-time existence of vortex sheets for the nonisentropic compressible Euler equations in two spatial dimensions, based on a linear stability result. The proof of the resolution for this nonlinear problem follows from certain a priori tame estimates on the effective linear problem and a suitable Nash?Moser iteration scheme. |
| Tohru | Nakamura | Viscous shock wave and singular limit for hyperbolic systems with Cattaneo's law | In this talk, we consider large time behavior of solutions to scalar conservation laws with an artificial heat flux term. In the case where the heat flux is governed by Fourier's law, the equation is scalar viscous conservation laws. In this case, existence and asymptotic stability of one-dimensional viscous shock wave, which is a smooth traveling wave solution, have been studied. The main concern in the current talk is a $2 \times 2$ system of hyperbolic equations with relaxation which is derived by prescribing Cattaneo's law for the heat flux. We consider the one-dimensional Cauchy problem for the system of Cattaneo-type and show existence and asymptotic stability of viscous shock wave. We also obtain the convergence rate by utilizing the weighted energy method. By letting the relaxation time zero in the system of Cattaneo-type, the system is formally deduced to scalar viscous conservation laws of Fourier-type. This is a singular limit problem which occurs an initial layer. We also consider the singular limit problem associated with viscous shock waves. |
| Khai | Nguyen | ||
| Michael | Ndjinga | Strong convergence of nonlinear finite volume methods for linear hyperbolic systems | Unlike finite elements methods, finite volume methods are far from having a clear functional analytic setting allowing the proof of general convergence results. In [4], compactness methods were used to derive convergence results for the Laplace equation on fairly general meshes. The weak convergence of nonlinear finite volume methods for linear hy- perbolic systems was proven in [5] using the Banach-Alaoglu compactness theorem. It allowed the use of general L 2 initial data which is consis- tent with the continuous theory based on the L 2 Fourier transform [1]. To our knowledge this was the first convergence results applicable to non differentiable initial data. However this weak convergence result seems not optimal with regard of numerical simulations. In this paper we prove that the convergence is indeed strong for a wide class of possibly nonlinear upwinding schemes. The context of our study being multidimensional, we cannot use the spaces L 1 and BV classically encountered in the study of 1D hyperbolic systems [2]. We propose instead the use of generalised p-variation function, ini- tially introduced by Wiener [6] and first studied by Young [7]. These spaces are compactly embedded in L p (see [9, 8]). They can therefore fit into the L 2 framework imposed by Brenner obstruction result [3]. Using estimates of the quadratic variation of the finite volume approximations we prove the compactness of the sequence of approximations and deduce the strong convergence of the numerical method. We finally discuss the applicability of this approach to nonlinear hyper- bolic systems. |
| Marko | Nedeljkov | Radially symmetric solutions to Euler system near the origin | A behavior of radially symmetric solutions to n-D compressible Euler gas dynamics model near the origin is not yet well understood. We do not know whether an incoming wave is reflected or accumulates in the origin as a some form resembling delta function (that means that it 'stays' at the origin). Using so called shadow wave solutions we succeeded to find out a partial answer to that question. A class of incoming waves partially reflects in a form of continuous wave but still making a kind of delta shock concentrated in the origin. We have proved that such solutions satisfies physically important inequalities - entropy condition is satisfied. The presentation is about joint work of the author and M. Oberguggenberger, L. Neumann (Univesity of Innsbruck) |
| Thi Hoai Thuong | NGUYEN | Stability of numerical schemes for the initial-boundary value problem of relaxation balance laws | In many industrial applications, models are based on hyperbolic partial differential equations which involve source terms. One of the main features of these models is related to the notion of dissipation, leading to smooth solutions and asymptotic stability. The most classical model is the Jin-Xin system [2] and we consider here the associated initial-boudary value problem (IBVP) in the quarter plane. This problem has been addressed by Xin and Xu who derive a necessary and sufficient condition for stiff stability in [1], i.e. stability uniformly with respect to the stiffness of the relaxation term. Our aim is to study the discrete case in the context of finite difference or finite volume approximations [2,3]. We study for instance the relationship between the stiff stability of numerical solutions and the Stiff Kreiss Condition (SKC) for the boundary problem. Due to the effects of the boundary layer and the interactions of the boundary and initial layer, numerical schemes have to be properly designed in order to provide accurate approximations and consistant behaviors. The asymptotic stability and boundary layer behavior are studied by discrete L2 analysis and energy estimate [3]. References [1] Zhouping Xin and Wen-Qing Xu. Stiff well-posedness and asymptotic convergence for a class of linear relaxation systems in a quarter plane. J. Differential Equations, 167(2):388437, 2000. [2] Shi Jin and Zhou Ping Xin. The relaxation schemes for systems of con- servation laws in arbitrary space dimensions. Comm. Pure Appl. Math., 48(3):235276, 1995. [3] B. Gustafsson, H.O. Kreiss, J. Oliger. Time dependent problems and difference methods, Wiley, 2013. |
| Guoxi | Ni | A conservative numerical method for compressible reactive fluids | In this work, we present a conservative interface method for reactive fluids problems, in which the standard finite volume scheme on Cartesian grids is modified by considering computational cells being cut by interface. While the discretized governing equations are updated conservatively, the method treats the topological changes naturally by combining interface description and geometric operations with a level set technique. Extensive tests in 1D are carried out, and 2D examples suggest that the present scheme is able to handle CJ problems in a straightforward way with good robustness and accuracy.. |
| Sebastian | Noelle | Accurate Front Velocities via New Space-Time Quadrature Rules | The occurrence of vacuum is a notorious difficulty in continuum mechanics, which affects the modeling, the mathematical analysis, and the design of stable numerical algorithms. When computing shallow water flows with conservative finite volume or discontinuous Galerkin schemes, infinitesimal quantities may occur in momentum and mass. The velocity is obtained by dividing the two, which is a De L'Hospital type limit. This often leads to erratic front velocities, which subsequently affect the CFL condition and hence the time step. We are currently developing a new space-time quadrature near the wet-dry front, for which we can guarantee accurate front-velocities which are within the physically reasonable bounds. This is joint work with Guoxian Chen, Yogiraj Mantri, Leonie Riebesam and Yulong Xing. |
| Helena | Nussenzveig Lopes | The Lighthill principle and vorticity estimates for incompressible flows with symmetry | In this talk we will discuss the vanishing viscosity limit for two classes of flows with symmetry: Plane-Parallel Channel Flows and Parallel Pipe Flows. We provide optimal rates of convergence of the Navier-Stokes solution towards the Euler solution and we investigate vorticity production on the boundary. In particular, we highlight the r?le of the Lighthill principle in obtaining vorticity estimates. |
| Víctor | Osores | An improved polydisperse sedimentation model and multilayer approximation with finite volume methods. | A two-dimensional vertical model to study polydisperse sedimentation with exact mass conservation is presented in this work. The model is discretized by a combination of a layer-averaged or multilayer technique and a finite volume method. The model allows to study the dispersion of solid particles of several species in a fluid, taking into account their diameters, concentrations, densities, their settling velocities and interaction between particles. The description is based on a modification of the well-known Maslliyah-Lockett-Bassoon (MLB) velocity functions. This modification is based on the definition of the average velocity of the mixture, and takes into account the densities of the solid particles, the fluid and the mixture. Usually, only the concentration of the solid particles is taking into account. In contrast to previous formulations we can now recover the exact mass conservation of the mixture. On the other hand, the multilayer or layer-averaged approach is deduced as a particular weak solution of the 2D vertical problem. This way, we obtain a horizontally one-dimensional system where the unknowns are the averaged velocities and concentrations on each layer, the transfer terms at each interface and the total mass. An explicit formula of the transfer terms is also deduced. Finally, we also derive an explicit bound of the eigenvalues of the system. This allows us to design an HLL-type path-conservative method to discretize the system. The HLL method of the full model is rewritten in terms of a HLL formulation by layers, which reduces the computational cost. In addiction, we apply this method to study several sedimentation problems. First, a comparison with classical polydisperse sedimentation models is presented for 1D vertical simulations. Secondly, we study the sedimention evolution in a type of basin. Finally, we test the robustness of the proposed numerical method by applying it to a scenario in which strong recirculations of the fluid are produced as a consequence of a sedimentation and interaction of partilces with the bottom. This contribution is based on joint work with Raimund B?rger (Universidad de Concepci?n, Chile) and Enrique D. Fern?ndez-Nieto (Universidad de Sevilla, Spain). |
| Lukas | Ostrowski | Compressible Droplet Impingement on Perfect and Rough Walls | We consider the impingement of a droplet onto a rough wall with high impact speed. In this situation compressible effects in the liquid play an important role. The compressed liquid leads to a shock wave which travels through the bulk and influences the dynamics heavily. Additionally, the microscopic structure of the rough wall has to be taken into account, since it also affects the spreading and splashing behavior of the droplet on the macroscale. For this purpose we consider a diffuse interface approach to model compressible two phase flow. Out of several phase field models which extend the classical compressible Navier--Stokes equations like Navier--Stokes--Korteweg or Navier--Stokes--Cahn--Hilliard systems, we use the following isothermal Navier--Stokes--Allen--Cahn model \cite{Dreyer2014}: \[ \begin{array}{cl} \begin{array}{rcccl} \partial_t \rho &+& \operatorname{div}(\rho \vec{v}) &=& 0, \\ \partial_t(\rho \vec{v}) &+& \operatorname{div}(\rho \vec{v}\otimes \vec{v}+{\mathbf{P}[\rho,\varphi]})&=&\operatorname{div}(\mathbf{S}), \\ \partial_t(\rho \varphi) &+& \operatorname{div}(\rho \varphi \vec{v}) &=& -\eta\left(\frac{\partial F}{\partial \varphi}-\gamma \Delta \varphi \right), \end{array} & \text{in } \mathbb{R}_{>0} \times \Omega \subset \mathbb{R}^d\\[6ex] \vec{v} = 0, \ \nabla \varphi \cdot \vec{n} = 0 &\text{in } \mathbb{R}_{>0} \times \partial \Omega. \end{array} \] Here $\rho>0$ denotes the fluid density and $\vec{v}$ the fluid velocity. The phase field parameter $\varphi \in [0,1]$ indicates the present phase. With $\mathbf{S}$ we denote the dissipative viscous part of the stress tensor and by $\eta>0$ the mobility. The non-dissipative part $\mathbf{P}$ of the stress tensor is defined as $\mathbf{P}[\rho,\varphi] = \left(-F+\rho\frac{\partial F}{\partial \rho}\right)\mathbf{I} + \gamma \nabla \varphi \otimes \nabla \varphi$. The free energy $F(\rho,\varphi,\nabla \varphi)$ consists of a double well, a combination of the free energies of each phase and a capillary term to account for capillary forces depending on the parameter $\gamma > 0$. We briefly discuss important properties of the model and present some numerical simulations on the macroscale. Due to the appearance of steep gradients and the convection dominated character of the flow Discontinuous Galerkin solver are favorable \cite{Bassi}. To this end we developed an energy consistent Discontinuous Galerkin solver. However, it is costly to resolve all microscopic structures in simulations. To overcome this issue homogenization and numerical upscaling techniques can be applied. We present an approach for numerical upscaling of the compressible Navier--Stokes--Allen--Cahn model in vicinity of rough surfaces and discuss differences to the steady or incompressible case. We apply techniques from \cite{Deolmi2017} to our two-phase situation. The idea is to perform simulations on a virtually smooth domain and include micro-scale effects through effective boundary conditions. To obtain these conditions cell problems corresponding to each roughness element have to be solved. |
| VIJAYARAJU | PALANIAPPAN | A Wavelet based Technique for Fractional-Order Hyperbolic Telegraphic Equation Via Operational Matrices | In this talk, a mathematical model is developed for the second order hyperbolic one-dimensional time fractional telegraph equation (TFTE). The fractional derivative is described in the Caputo sense. The governing equations are solved by a recent approximation method known as wavelet operational matrix (WOM) method. This method is a powerful mathematical technique for solving a wide range of problems. Using WOM method, the underlying TFTE is converted into a system of algebraic equations. Some numerical examples of TFTE are provided in order to check the eectiveness, accuracy and convergence of the method. The computed results are also depicted graphically. |
| Michele | Palladino | Growth Model of Tree Stems and Vines | In this talk, we propose a model describing the growth of tree stems and vine, taking into account also the presence of external obstacles. The system evolution is described by an integral differential equation which becomes discontinuous when the stem hits the obstacle. The stem feels the obstacle reaction not just at the tip, but along the whole stem. This fact represents one of the main challenges to overcome, since it produces a cone of possible reactions which is not normal with respect to the obstacle. However, using the geometric structure of the problem and optimal control tools, we are able to prove the well-posedness of the model under natural assumptions on the initial data. Further research directions to model different natural shapes will be also presented. |
| Carlos | Pares | High-order well-balanced numerical methods for Euler equations | The goal of this work is to design high-order well-balanced numerical methods for the 1d compressible Euler equations with a gravitational source term. More precisely, we focus on the design of numerical methods that preserve all the stationary solutions of a family characterized by (i) the Equation of State; (ii) the hydrostatic pressure law: ∂p/∂x=-ρΦ^' (x), where ρ is the density and Φ is the gravitational potential; and (iii) a given functional relationship between the thermodynamic variables, for instance: ρ=f(θ), e=g(θ). where θ is a known variable and f,g two given functions. Some known families of stationary solutions, such as polytropic, isentropic, or isothermal solutions for different EoS, can be written in this form. The numerical methods presented here are based on standard reconstruction operators of arbitrary accuracy. According to [1], given a family of cell averages {U_i}, the reconstruction function at the ith cell I_i is computed as follows: (i)Look for the stationary solution U_i^* (x) of the family to be preserved whose cell-average is equal to U_i. (ii)Compute the fluctuations V_j=U_j-1/Δx ∫_(I_j)▒〖U_i^* (x)dx〗 in the cell I_j belonging to the stencil S_i of the ith cell. (iii)Apply the reconstruction operator to {V_j }_(j∈S_i) to obtain Q_i (x). (iv)Define the reconstruction function at I_i by P_i (x)=U_i^* (x)+Q_i (x) The well-balanced property of the numerical methods will be shown and illustrated in different test problems. The implementation difficulties will be discussed as well. |
| Carlos | Pares Pulido | High-order, essentially non-oscillatory, positivity preserving methods for incompressible flows on staggered meshes | This talk is based on joint work with S. Mishra (D-MATH, ETH Zurich), K. Pressel, T. Schneider (Climate Dynamics Group, Caltech), and C.-W. Shu (Division of Applied Mathematics, Brown University). We will present a review of different high-order accurate (essentially) non-oscillatory finite difference/volume schemes for the incompressible Euler equations in both two and three space dimensions, with a focus on positivity preservation. These equations are the fundamental governing equations for atmospheric flows and their numerical approximation is an essential component of all modern climate dynamics codes. Several of these codes use very high-order finite difference schemes, based on central stencils to solve these equations (e.g. Wicker-Skamarock, 2001; Morinishi, 1998). However, these central schemes suffer from several defects; namely, they can produce spurious oscillations when the solution contains sharp gradients, such as in shear layers, and they are often in practice bounded to second order accuracy. Given these limitations, WENO schemes have been recently proposed as the suitable simulation framework for some atmospheric flows (Pressel et al., 2015). A properly high-order scheme can be developed, however it suffers from a lack of positivity. For simulation of atmospheric flows, even under the incompressible approximation, positivity is required of many scalar variables, like water content. In this talk, we will discuss the possibility of obtaining a high-order, positivity-preserving advection scheme for scalars, and review approaches to its design, both existing and newly developed. |
| Marica | Pelanti | On the wave structure similarity of the HLLC and Roe Riemann solvers. Application to low Mach number preconditioning for single-phase and two-phase flows | Finite volume Godunov-type schemes based on Riemann solvers are widely used to compute solutions to hyperbolic systems of equations, in particular in the context of fluid dynamics. Some of the most popular approximate Riemann solvers are the celebrated solver of Roe (1981) and the solver of Harten--Lax--van Leer (HLL) (1983) and its variants, such as the HLLEM and the HLLC solvers. The HLLC solver (HLL with Contact restoration) introduced by Toro, Spruce and Speares (1994) for the Euler equations of gas dynamics has especially gained increasing popularity over the last two decades since it possesses some of the good properties of the Roe solver and in addition it satisfies important entropy and positivity conditions with no need of special fixes. In the present work we rewrite the classical HLLC Riemann solver in a novel form that highlights the formal mathematical similarity of its wave structure with the one of the Roe solver. In particular, we show by very simple algebraic manipulations that the acoustic waves of the HLLC solver, similarly to the Roe solver, are collinear to vectors corresponding to the eigenvectors of the Euler equations associated to the acoustic fields evaluated in a special state, which is a function of the left and right Riemann data. The strengths of these waves have also expressions in terms of jumps in pressure and normal velocity formally analogous to those of the Roe solver. The revealed mathematical similarity of the wave structure of the two solvers might be useful to extend to the HLLC method some numerical techniques devised specifically for the Roe's method. As an example of application we illustrate the design and the analysis of a Turkel-type low Mach number preconditioning technique for the HLLC scheme applied to the Euler equations, by mimicking methodologies proposed in the literature for the Roe's scheme [1]. An extension of this low Mach number HLLC-Turkel method to a hyperbolic six-equation two-phase flow model is also presented [2], and numerical results are shown for both single-phase and two-phase liquid-gas flows test problems in two spatial dimensions. Other potential benefits of the novel reformulation of the HLLC solver are briefly discussed. [1] H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit, Computers and Fluids 28 (1999), 63-86. [2] M. Pelanti, Low Mach number preconditioning techniques for Roe-type and HLLC-type methods for a two-phase compressible flow model, Appl. Math. Comp., 310 (2017), 112-133. |
| Teddy | Pichard | A system of hyperbolic PDE with a discontinuous source term | This work is motivated by liquid-vapor modeling with boiling effects. A model of Drift-flux type is considered for the motion of the homogenized flow (liquid and vapor together). This model is represented by a system of hyperbolic PDE with a source term that is defined as a discontinuous function of the unknowns. This discontinuity models the change from liquid to vapor in the flow when a certain enthalpy threshold is reached. We first analyse the existence of steady states in this model. Then we study the existence of unsteady solution and their structure for a simplified system. Finally based on this structure, we construct non-oscillatory numerical schemes for such a system. |
| Bojan | Popov | Invariant domain preserving approximation of the Euler equations via convex limiting | A second-order finite-element-based method for approximating the compressible Euler equations is introduced. The method preserves all the known invariant domains of the Euler system: positivity of the density, positivity of the internal energy and the local minimum principle on the specific entropy. In addition, it preserves the upper bound on the density if the equation of state allows for it. The technique combines a first-order, invariant domain preserving, Guaranteed Maximum Speed method using a Graph Viscosity (GMS-GV1) with an invariant domain entropy consistent, possibly violating the invariant domain, high-order method. Invariant domain preserving auxiliary states, naturally produced by the GMS-GV1 method, are used to define local bounds for the high-order method which is then made invariant domain preserving via a convex limiting process. Numerical tests confirm the second-order accuracy of the new GMS-GV2 method in the maximum norm, where 2 stands for second-order. The proposed convex limiting is generic and can be applied to other approximation techniques and other hyperbolic systems with convex invariant domains. |
| Corentin | Prigent | A kinetic approach to the bi-temperature Euler model | The aim of this work is the study of out-of-equilibirum plasma physics. It is a multiscale problem involving both very small lengths (Debye length) and high-frequency oscillations (electronic plasma frequency). Transport of charged particles (electrons and ions) in context of Inertial Confinement Fusion (ICF) can be modelled by the bi-temperature Euler equations, which are a non-conservative hyperbolic system. It contains so-called non-conservatives terms, which cannot be put in divergential form. Such terms are not well-defined, and, in the situation of shocks, computing exact or approximated solutions is a challenging issue. The bi-temperature Euler model can be recovered using a Chapman-Enskog expansion from an underlying kinetic approach of this system, the Vlasov-BGK-Ampere system, which is conservative. We are interested in the numerical resolution of this kinetic model, in a macroscopic setting. Hence, a scaling is performed on this model in order to exhibit the behaviour of the system in large scale configurations. The major issue of such a system is that the Maxwell equations are describing small scale electromagnetics. At the macroscopic level, these equations degenerate into algebraic relations, preventing their use for computation purposes. Hence, we derive an Asymptotic-Preserving numerical method, which is able to solve the system even when these small scales (Debye length, electronic plasma frequency) are not resolved. Numerical test cases are studied. Several well-known Riemann problems are solved with our method and then compared with methods for the macroscopic bi-temperature Euler model. |
| Emanuela | Radici | Deterministic particle approximation for scalar aggregation-diffusion equations with nonlinear mobility | In this work we would like to describe the one dimensional dynamic of a biological population influenced by the presence of a nonlocal attractive potential and a diffusive term, under the constraint that no over crowding can occur. This setting can be expressed by a class of aggregation-diffusion PDEs with nonlinear mobility. We investigate the existence of weak type solutions obtained as large particle limit of a suitable nonlocal version of the follow-the-leader scheme, which is interpreted as the discrete Lagrangian approximation of the target continuity equation. We restrict the analysis to nonnegative bounded initial with finite total variation, away from vacuum and supported in a closed interval with zero-velocity boundary conditions. The main novelties of this work concern the presence of a nonlinear mobility term and the non strict monotonicity of the diffusion function, thus, our result applies also to strongly degenerate diffusion equations. We also address the pure attractive regime, where we are able to achieve a stronger notion of solution. Indeed, in this case our scheme converges towards the unique entropy solution to the target PDE as the number of particles tends to infinity. This is a joint work with Marco Di Francesco and Simone Fagioli. |
| Bogdan | Radu | A Mixed FEM with mass lumping for acoustic wave propagation | We consider the numerical approximation of acoustic wave propagation by a mixed finite element method based on the BDM1-P0 finite element. A mass-lumping strategy is utilized to enable an efficient time discretization. Due to mass lumping, the approximation order of the mixed method is reduced to first order. We show that the numerical approximation still carries second order information and propose a post-processing strategy for both variables which yields piecewise linear approximations of second order. Numerical tests are presented to illustrate the efficiency of the proposed approach. Acknowledgement : This work is supported by the "Excellence Initiative" of the German Federal and State Governments and the Graduate School of Computational Engineering at the Technische Universität Darmstadt. |
| Audun | Reigstad | A regularized system for the nonlinear variational wave equation | We present a generalization of the nonlinear variational wave (NVW) equation, which is given by \begin{align*} u_{tt}-c(u)(c(u)u_x)_x&=-\frac{c'(u)}{4}(\rho^2 + \sigma^2),\\ \rho_t - (c(u)\rho)_x &= 0,\\ \sigma_t + (c(u)\sigma)_x &=0. \end{align*} The motivation for the derivation of this generalization has been the Camassa--Holm (CH) equation, for which the two-component CH system provides a generalization which makes it possible to approximate each conservative solution of the CH equation by smooth solutions to the two-component CH system. In particular, there exist solutions, which if smooth on some interval initially, remain smooth by following the interval along characteristics. We are going to show that solutions to the above generalization of the NVW equation, which are initially smooth on a finite interval and satisfy $\rho(0,x)>0$ and $\sigma(0,x)>0$ on this interval, will remain locally smooth. Furthermore, we prove that by taking $\rho_{n},\sigma_{n}\rightarrow 0$ with $\rho_{n},\sigma_{n}>0$, the corresponding smooth solutions converge to the conservative solution of the NVW equation. This provides a new way of studying conservative solutions. This is joint work with K. Grunert, H. Holden, and X. Raynaud. |
| Florent | Renac | Adjoint approximation of a nonlinear hyperbolic system in non-conservation form | We consider the approximation of adjoint-based derivatives for discontinuous solutions of an unsteady one-dimensional nonlinear hyperbolic $2\times2$ system. The system possesses a genuinely nonlinear field and a linearly degenerate (LD) field associated to the nonconservative product. This model is representative for instance of two-phase flow problems where the LD field plays the role of interface velocity [1]. We first characterize solutions of the Riemann problems for this system and construct a first-order entropy satisfying finite volume discretization allowing convergence to the physically relevant solution. Then, we derive the adjoint equations in strong form of the hyperbolic system and, for a discontinuous primal solution, we characterize the associated jump relations across a discontuity. The novelty is that the adjoint solution may be discontinuous across the discontinuity in contrast to the case of nonlinear scalar hyperbolic equations with a convex flux [2]. Furthermore, the discrete adjoint solution from the finite volume approximation is shown to be consistent with the strong adjoint solution. Numerical experiments for the approximation of both direct and adjoint problems will be presented to assess the performances of the present method. [1] F. Coquel, T. Gallou?t, J.-M. H?rard and N. Seguin, Closure Laws for a Two-Fluid Two-Pressure Model, C. R. Acad. Sci. Paris, 334 (2002), pp. 927?932. [2] S. Ulbrich, A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms, SIAM J. Control Optim., 41 (2002), pp. 740?797. |
| Johanna | Ridder | Global solution of the Riemann problem for a nonstrictly hyperbolic 3x3 system of conservation laws modeling polymer flooding with gravity and adsorption | We construct a global solution for the Riemann problem of a 3 ? 3 system of conservation laws that arises in the modeling of oil reservoirs where oil in the rock is replaced by a polymer-water mixture. This model is an extension of the classical polymer problem analyzed by Isaacson and Temple in 1981/82 and consequent work. However, our model differs from previous work in that it includes effects of gravity and adsorption and at the same time allows for discontinuities in the permeability of the rock. The main difficulty of this type of models is that resonance can occur, i.e., eigenvalues of different families can coincide. In our case, this happens along two distinct lines in the phase plane of the reduced 2 ? 2 system when permeability is assumed to be constant. In addition, there are contact discontinuities along the lines where the flux vanishes, and the discontinuity of the permeability. A careful analysis allows us to first construct a global solution of the Riemann problem for the reduced 2 ? 2 system. We then use this solution together with known results about conservation laws with discontinuous flux functions to solve the Riemann problem for the full 3 ? 3 system where the permeability may have a fixed discontinuity. |
| Daniel | Rodriguez Marroquin | Short Wave-Long Wave Interactions in Planar Magnetohydrodynamics | We study several mathematical aspects of a system of equations modelling the interaction between short waves, described by a nonlinear Schr?dinger equation, and long waves, described by the equations of magnetohydrodynamics (MHD) for a compressible, heat conductive fluid. The system in question models a phenomenon like that of the aurora, also known as polar lights, where a short wave propagates along the streamlines of a magnetohydrodynamic medium. We focus on the one dimensional (planar) version of the model and address the problem of well posedness as well as convergence of the sequence of solutions as the bulk viscosity tends to zero together with some other interaction parameters. |
| Elena | Rossi | Recent results on conservation laws in bounded domains | In this talk, we present a unique general framework for various definitions of solution to initial boundary value problems (IBVPs) for balance laws. We compare how boundary conditions are fulfilled in each case, providing the assumptions ensuring the equivalence among the definitions considered. We extend to this setting the basic well posedness results found in the literature, obtaining in particular a sharp estimate for the dependence of the solution on both the initial and boundary data. Concerning the stability with respect to the flux, we obtain it in the case of a linear conservation law with homogeneous boundary conditions. This allows to prove the well posedness of a class of systems of non local conservation laws in a bounded domain. In the one dimensional case with a time dependent flux, we prove the well posedness and the stability of solutions to the IBVP with respect to the initial datum, to the boundary datum and to the flux, both on an interval and on a half line. This work is in collaboration with Rinaldo M. Colombo (University of Brescia, Italy). |
| Adrian Montgomery | Ruf | The Ostrovsky?Hunter equation with Dirichlet boundary conditions | We prove convergence of a finite difference scheme to the unique entropy solution of a general form of the Ostrovsky?Hunter equation on a bounded domain with non-homogeneous Dirichlet boundary conditions. Our scheme is an extension of monotone schemes for conservation laws to the equation at hand. The convergence result at the center of this work also proves existence of entropy solutions for the initial-boundary value problem for the general Ostrovsky?Hunter equation. Additionally, we show uniqueness using Kruzkov?s doubling of variables technique. We also include numerical examples to confirm the convergence results and determine rates of convergence experimentally. This is a joint work with Johanna Ridder. |
| Smita | Sahu | On coupled schemes for advection and Hamilton-Jacobi-Bellman equations | Our goal is to study some numerical approximations of particular advection and Hamilton-Jacobi-Bellman equations in dimension 1 and with possibly discontinuous initial data. The coupled scheme proposed in [3] is general and can in principle be applied to couple many different schemes, for example, one can couple an accurate method well adapted where the solution is smooth with another method designed to treat discontinuities and/ or jumps in the gradients. One has to decide where to apply the first or the second method and this is done by means of a switching parameter which must be computed in every cell at every time step. We investigate in particular, coupling between an ultra-bee [1] and a semi-Lagrangian [2] scheme. We prove the coupled scheme is $L^\infty$ stable, consistent and TVB for the advection equation. We also provide numerical examples to validate their accuracy in solving smooth and discontinuous solutions. \textbf{References}\\ $[1]$ O. Bokanowski and H. Zidani, Anti-dissipative schemes for advection and application to HJB equations, {\emph J. Scient. Comput.} 30(1) pp. 1-33,2007.\\ $[2]$ M. Falcone and R. Ferretti, Discrete time high-order scheme for viscosity solutions of Hamilton-Jacobi equation, {\emph Numer. Math.}, vol. 67, p. 315, 2014. $[3]$ S. Sahu, Coupled scheme for linear and non-linear Partial differential equations. To appear Springer Proceedings in Mathematics and Statistics. 2017. |
| Océane | Saincir | Numerical approximation of a radiative transfer model for the description of astrophysical flows | In this work, we define a new numerical strategy for the simulation of astrophysical flows in the diffusion limit, where the material is assumed to be opaque. It is well-known that in the general case, the full system, involving both the classical hydrodynamical contribution and the equations governing the radiative quantities (such as the radiative flux and the radiative energy) requires a huge computational effort when dealing with numerical simulations. In our context, the expression of radiative flux can be simplified which gives a diffusion term in the equation governing the radiative energy. We first present the diffusion approximation model and then describe our algorithm that consists in handling the hyperbolic part with a finite volume resolution based on HLL solvers and the other terms with an implicit method, that leads us to moderate CPU time. Finally, numerical simulations are presented in order to demonstrate the efficiency and accuracy of our method, performing well-known tests in the astrophysical literature. In particular, supercritical and subcritical radiative shocks are observed, in agreement with previous methods. A ?shock-tube like? benchmark in the radiative case is also presented. |
| Khaled | Saleh | A class of asymptotic preserving numerical schemes for low Mach number flows. | Since seminal papers published in the middle of the sixties, low-order staggered schemes for incompressible flow computations have received a considerable attention. The staggered discretization is a space structured or unstructured discretization where the scalar unknowns (the pressure) are located at the cell centers while the vector unknowns (the velocity) are located at the cells faces. This interest is essentially motivated by the fact that they combine a low computational cost with the so-called inf-sup or LB stability condition, which prevents from the odd-even decoupling of the pressure. For several years, an important effort has been dedicated to the extension of staggered numerical schemes for the approximation of compressible flows. Usually expected properties have been proven for these new schemes: existence of the numerical solution, preservation of the admissible phase space, stability, entropy inequalities, consistency in the sens of Lax and Wendroff. Besides their efficiency and simplicity (the numerical fluxes are easily computed), these schemes have the following remarkable property: if the numerical density is constant, they boil down to the standard staggered algorithm for incompressible flows, which is known to be stable and efficient. In particular, as for the incompressible case, the staggered discretization ensures the control of the L^2 norm of the pressure through of the control of the H^{-1} norm of its gradient (thanks to the inf-sup stability condition). In this talk, I will focus on the barotropic compressible Navier-Stokes equations. I will prove that, as the Mach number tends to zero, the solution of the implicit staggered scheme for these equations converges towards the solution of the standard staggered scheme for the incompressible Navier-Stokes equations. In particular, the numerical density tends towards a constant as the Mach number tends to zero. Such a result follows from a similar analysis to that of Lions and Masmoudi (1998) at the continuous level for weak solutions of the barotropic compressible Navier-Stokes equations. It extends to other time discretizations such as the so called pressure-correction scheme (adapted to compressible models). It is these schemes that are used in practice in the industrial codes such as P2REMICS. A code developed by the IRSN (french Institut for Radioprotection and Nuclear Safety) for the simulation of deflagrations. |
| Najmeh | Salehi | OFF-RAMP COUPLING CONDITIONS DEVOID OF SPURIOUS BLOCKING AND RE-ROUTING | When modeling vehicular traffic flow on highway networks via macroscopic models, suitable coupling conditions at the network nodes are crucial. Frequently, the evolution of traffic flow on each network edge is described in a lane-averaged fashion using a single-class Lighthill-Whitham-Richards model. At off-ramps, split ratios (i.e., what percentage of traffic exits the highway) are prescribed that can be drawn from historic data. In this situation, classical FIFO coupling conditions yield unrealistic results, in that a clogged off-ramp yields zero flux through the node. As a remedy, non-FIFO conditions have been proposed. However, as we demonstrate here, those lead to spurious re-routing of vehicles. Then, a new coupling model, FIFOQ, is presented that preserves the desirable properties of non-FIFO models, while not leading to any spurious re-routing. |
| Saurav | Samantaray | Asymptotic Preserving Time Integrators for Low Mach Number Limits of the Euler Equations with Gravity . | We consider the compressible Euler equations in a gravitational field. After scaling, the Euler system gives rise to two non-dimensional, characteristic numbers: the Mach number and the Froude number. It is well known that when the Mach number goes to zero, the Euler equations converge to their incompressible counter part. However, when the Froude number is also small, under appropriate scaling, many other limit equations, such as the Bousinesq equations, Anelastic equations etc. can also be derived by a systematic asymptotic analysis. All these models are collectively called sound proof models and their numerical resolution is very challenging due to stability constraints, order reduction etc. .We propose a generic framework for the numerical solution of various sound proof models.The main idea behind our numerical method is a semi-implicit discretisation of the flux and source terms via IMEX-RK time integrators. The explicit part of the scheme is treated by standard Godunov- Type finite volume schemes, and the implicit part of the scheme is converted to the solution of an elliptic problem. The overall scheme overcomes timestep restrictions and is accurate for the particular limit system under consideration. Numerical experiments conducted demonstrate the theoretical propositions. |
| Steven | Schochet | Three-Scale Singular Limits of Evolutionary PDEs | Singular limits of a class of evolutionary systems of partial differential equations having two small parameters and hence three time scales are considered. Under appropriate conditions solutions are shown to exist and remain uniformly bounded for a fixed time as the two parameters tend to zero at different rates. A simple example shows the necessity of those conditions in order for uniform bounds to hold. Under further conditions the solutions of the original system tend to solutions of a limit equation as the parameters tend to zero. Joint work with Bin Cheng and Qiangchang Ju |
| Matthew | Schrecker | Compressible Euler equations with geometric effects | In this talk, I will present some recent results concerning the existence of globally-defined entropy solutions to the 1D compressible, isentropic Euler equations with geometric effects. The types of effect will typically be those arising from transonic nozzle flow or spherical symmetry. I will present a robust framework for demonstrating the existence of entropy solutions to these equations via a carefully constructed viscous approximation and compactness argument. This is joint work with Gui-Qiang Chen. |
| Christian | Schulze | Cellular mixing with bounded palenstrophy | We study the problem of optimal mixing of a passive scalar advected by an incompressible flow on the two dimensional unit square. The scalar solves the continuity equation with a divergence-free velocity field with uniform-in-time bounds on the palenstrophy. For velocity fields with the above constraint, it is known that the decay of both the geometric as well as the functional mixing scale cannot be faster than exponential. Numerical simulations suggest that this exponential lower bound is in fact sharp, but so far there is no explicit analytical example which matches this result. We analyze velocity fields of cellular type, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially. |
| Jingyang | Shu | Sharp Fronts for the SQG Equation | Temperature discontinuities in the Surface Quasi-Geostrophic (SQG) equations support surface waves. Weakly nonlinear surface waves on SQG fronts can be described by expansion of suitably regularized contour dynamics equations. For fronts that are described as a graph, we prove short-time well-posedness results for a nonlocal, cubically nonlinear, approximate equation with logarithmic dispersion. We also derive a dispersive estimates. This is joint work with John K. Hunter and Qingtian Zhang. |
| Paramjeet | Singh | High-order finite volume approximation based on IMEX-WENO for nonlinear age-structured population model | The age-structured population models play a key role in population dynamics. The population of human, plant, and animal all are age-structured. Age is an intrinsic and main argument of structuring a population. Many of the internal variables depend on the age because the difference in ages means the difference in reproduction, the difference in survival capacities, and different behaviors. We consider a finite maximum age and the resulting equation is a hyperbolic conservation laws. We design a high-order implicit-explicit weighted non-oscillatory (IMEX-WENO) scheme for the solution of the population age density model with the nonlinear mortality rate as well as fertility rate. The mortality rate is the combination of natural mortality that comes to be unbounded at maximum age and bounded external mortality rate that includes external resources as well as seasonality. The existence of global terms in the mortality function as well as in the boundary condition is the main technical complication which provides high nonlinearity for the model equation. The proposed scheme is capable to maintain the high-order accuracy in the global terms. We show the performance of the newly developed scheme in a series of examples. |
| Susanne | Solem | A second-order numerical method for the aggregation equation | Inspired by TVD limiter-based second-order schemes for conservation laws, we derive a second-order accurate numerical method for multi-dimensional aggregation equations. The motivation for the scheme stems from an equivalence between a one-dimensional aggregation equation and a Burgers-type equation proved by Bonaschi et al. [ESAIM: COCV, 21 (2015)]. The scheme is validated by a proof of convergence in the 1--Wasserstein metric and a selection of numerical experiments. This is joint work with J. A. Carrillo and U. S. Fjordholm. |
| Stefano | Spirito | Quasiconvex elastodynamics: weak-strong uniqueness for measure-valued solutions | This talk concerns a classical system of conservation laws arising in elasticity. One of the main difficulty in elastodynamics is that convex stored energy functions are in contradiction with the basic physical principle of frame-indifference. In this talk I will present a recent result, obtained in collaboration with Kostantinos Koumatos, University of Sussex (UK), proving the weak-strong uniqueness of measure-valued solutions under the assumption that the stored-energy function is strongly quasiconvex. The proof combines the relative entropy method and several tools borrowed from the calculus of variations. |
| Varsha | Srivastava | Robust Numerical Method for Time-Dependent Singularly Perturbed Semilinear Problems | In solving time-dependent singularly perturbed semilinear problems by the standard finite differences or finite element methods, the corresponding discrete problem on each time level is formulated as a nonlinear systems of algebraic equations; and to solve these nonlinear systems of algebraic equations, we require some iterative method for the computation of the numerical solutions. In this article, a finite difference numerical method is used to solve the time-dependent singularly perturbed semilinear convection-diffusion problems. The numerical approximations to the solution are generated using a backward Euler method in time and a HODIE method in space via simultaneous discretization. The stability for the present time-dependent semilinear problems (both continuous and discrete) are proved by the inverse-monotonicity properties of the classes of linear initial-boundary value problems. The given method is shown to have first order parameter-uniform convergence in time and almost second order parameter-uniform convergence in space. Numerical result is given to support the theoretical error bounds of the numerical method. |
| Qing | SUN | ||
| Wenjun | Sun | Multiscale Radiative Transfer in Cylindrical Coordinates | The radiative transfer equations in cylindrical coordinates are important in the application of inertial confinement fusion. In comparison with the equations in Cartesian coordinates, an additional angular derivative term appears in the cylindrical case. This term adds great difficulty for a numerical scheme to keep the conservation of total energy. In this report, based on weighting factors, the angular derivative term is properly discretized, and the interface fluxes in the radial r-direction depend on such a discretization as well. A unified gas kinetic scheme (UGKS) with asymptotic preserving property for the gray radiative transfer equations is constructed in cylindrical coordinates. The current UGKS can naturally capture the radiation diffusion solution in the optically thick regime with the cell size being much larger than photon's mean free path. At the same time, the current UGKS can present accurate solutions in the optically thin regime as well. Moreover, it is a finite volume method with total energy conservation. Due to the scale dependent time evolution solution for the interface flux evaluation, the scheme can cover multiscale transport mechanism seamlessly. The cylindrical hohlraum tests in inertial confinement fusion are used to validate the current approach, and the solutions are compared with implicit Monte Carlo result. |
| Masahiro | Suzuki | Bifurcation analysis of an equation for gas discharge | The main purpose of this talk is to analyze mathematically the fundamental mechanism of gas discharge. Specifically, we study an initial boundary value problem to the Morrow model over a bounded interval. This problem has a trivial stationary solution for any voltage, which is adopted as a boundary condition. We first show that there exists a threshold of voltage at which the trivial solution becomes from stable to unstable. This threshold can be understood as a sparking voltage for the happening of gas discharge. Furthermore, by regarding the voltage as the bifurcation parameter, we prove that a non-trivial solution bifurcates from the trivial solution. |
| Eitan | Tadmor | Flocking Hydrodynamics | I will describe recent works on the regularity and large-time flocking behavior of one- and multi-species dynamics driven by alignment. |
| Yuka | Teramoto | On the spectrum for artificial compressible system | This talk is concerned with stability of stationary solutions of the incompressible Navier?Stokes system and the corresponding artificial compressible system. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero Mach number limit which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small Mach number. The result is applied to the Taylor problem. |
| Ignacio | Tomas | Invariant-domain preserving entropy-stable schemes: discretization independent framework. | The first step in the development of a high-order scheme is the development of a robust first-order method (ideally) endowed with a sound theoretical basis. With that goal in mind, we develop a general framework of first-order fully-discrete numerical schemes that are guaranteed to preserve every convex invariant of the hyperbolic system and satisfy every entropy inequality. This framework is not tied to any particular space discretization technique (e.g. finite volumes, continuous or discontinuous finite elements) and it is of general applicability: it works in any mesh, in any space dimension, and uses neither exact nor approximate Riemann solvers. One of the by-products of this presentation is the first (to the best of our knowledge) fully-discrete entropy-stable dG scheme for hyperbolic systems. This talk is the first-one from a sequel of four: the other three talks presented by Jean-Luc Guermond, Yuchen Hua, and Bojan Popov. |
| Davide | Torlo | Asymptotic Preserving Deferred Correction Residual Distribution schemes | Davide Torlo, joint work with R?mi Abgrall The AP DeC RD schemes are a class of schemes that can be used to solve relaxed hyperbolic systems. In particular, we focused on kinetic schemes, but it can be extended to multiphase flows, viscoelasticity or relaxing gas flows and so on. The main ingredient of these models is a stiff relaxation term leaded by a relaxation parameter. The method is capable to catch the correct behaviour in both regimes: small and big relaxation parameter. Moreover, we proved that the method can preserve the analytic limit as the relaxation parameter goes to 0. (in preparation, D.T. and R.A., Asymptotic Preserving Deferred Correction scheme.) The scheme is composed of several procedures. First of all, we need a special treatment of the source relaxation term, since it may be stiff. So we adopted an IMEX scheme, implicit in source and explicit in the advection term. In this way, we do not need to use very small timesteps for the scheme to be stable. (S. Boscarino, L. Pareschi and G. Russo. (2011). Implicit-Explicit Runge--Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit. SIAM Journal on Scientific Computing.) In order to get high order schemes, we used residual distribution (RD) framework to do the spatial discretization. It is a finite elements based scheme, easy to code, that can get high order, with a compact stencil. Moreover, it can recast other well known finite element or finite volume schemes. (M. Ricchiuto and R. Abgrall. Explicit Runge-Kutta residual distribution schemes for time dependent problems: Second order case. J. Comput. Phys. 229 (16) (2010): 5653 - 5691.) For the time discretization we use the Deferred Correction (DeC) method, which is a way to combine 2 methods: the first, which is stable and low order, and the second one, more difficult to solve, but high order. This iterative method can reach the high order with few steps, solving implicitly only the stable scheme. (A. Dutt, L. Greengard and V. Rokhlin. Spectral Deferred Correction Methods for Ordinary Differential Equations. BIT Numerical Mathematics. 40 (2) (2000): 241-266.) Thanks to the DeC method, we can write a first order IMEX approximation where the implicitness are confined to diagonal terms in the mass matrix, so that we don't really need to recast implicit methods to solve the scheme. Indeed, the final method will be completely explicit. To show the performance of our scheme, we tested it on some test cases in 1D and 2D. |
| Kazuyuki | Tsuda | Diffusion wave property and smoothing effect of solutions to the compressible Navier-Stokes-Korteweg system | Time decay estimate of solutions to the compressible Navier-Stokes-Korteweg system is studied. Concerning the linearized problem, the decay estimate with diffusion wave property for an initial date is derived from hyperbolic aspect of the system. As an application, the time decay estimate of solutions to the nonlinear problem is given. In contrast to the compressible Navier-Stokes system, for linear system regularities of the initial date are lower and independent of the order of derivative of solutions owing to smoothing effect from the Korteweg tensor. Furthermore, for the nonlinear system diffusion wave property is obtained with an initial date having lower regularity than that of studies of the compressible Navier-Stokes system. |
| Naoki | Tsuge | GLOBAL ENTROPY SOLUTIONS TO THE COMPRESSIBLE EULER EQUATIONS IN THE ISENTROPIC NOZZLE FLOW | We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle ow is closely related to astrophysics.These phenomena are governed by the compressible Euler equation, which is one of crucial equations in inhomogeneous conservation laws. In this paper, we consider its unsteady ow and devote to proving the global existence and stability of solutions to the Cauchy problem for the general nozzle. Although the subject is important in Mathematics, Physics and engineering, it remained open for a long time. The problem seems to lie in a bounded estimate of approximate solutions, because we have only method to investigate the behavior with respect to the time variable. To solve this, we first introduce a generalized invariant region. Compared with the existing ones, its upper and lower bounds are extended constants to functions of the space variable. However, we cannot apply the new invariant region to the traditional difference method. Therefore, we invent the modified Godunov scheme. The approximate solutions consist of some functions corresponding to the upper and lower bounds of the invariant regions. These methods enable us to investigate the behavior of approximate solutions with respect to the space variable. The ideas are also applicable to other nonlinear problems involving similar difficulties. |
| Yong | WANG | Well-posedness on a Generalized Poisson-Nernst-Planck-Navier-Stokes Equations | We use an energetic variational approach to derive a new hydrodynamic model, which could be called a generalized Poisson-Nernst-Planck-Navier-Stokes system. Such the system could describe the dynamics of the compressible conductive fluid with the dilute/crowded charged particles and be used to analyze the interactions between the macroscopic fluid motion and the microscopic charge transportation. Then, we develop a general method to obtain the unique local classical solution, the unique global solution under small perturbations and the optimal decay rates of the solution and its derivatives of any order. |
| Lei | Wang | Existence and stability of stationary solution to compressible Navier-Stokes-Poisson equations in half line | In this talk, we investigate the asymptotic stability of the stationary solution to the outflow problem for the compressible Navier-Stokes-Poisson system in a half line. We show the existence of the stationary solution with the aid of the stable manifold theory. The time asymptotic stability of the stationary solution is obtained by the elementary energy method. Furthermore, for the supersonic flow at spatial infinity, we also obtain an algebraic and an exponential decay rate, when the initial perturbation belongs to the corresponding weighted Sobolev space. The proof is based on a time and space weighted energy method. |
| Yi | Wang | Stability of basic wave patterns for some kinetic equations | First, I will talk about the hydrodynamic limit of the Boltzmann equation to the compressible Euler equations in the setting of 1D generic Riemann solutions, which is the superposition of three basic wave patterns to Euler equations, i.e., shock wave, rarefaction wave and contact discontinuity. Then I will show the nonlinear stability of these three basic wave patterns to the bipolar Vlasov-Poisson-Boltzmann system based on a new micro-macro type decomposition around the local Maxwellian to the system and our recent result on the time-asymptotic stability of planar rarefaction wave to 3D Boltzmann equation. |
| Tianyi | Wang | On Steady Euler Flows with Large Vorticity and Characteristic Discontinuities | In this talk, we want to present the existence and uniqueness of smooth solutions with large vorticity and weak solutions with vortex sheets/entropy waves for the steady Euler equations for both compressible and incompressible fluids in arbitrary infinitely long nozzles. A new approach is introduced for the existence of smooth solutions without assumptions on the sign of the second derivatives of the horizontal velocity, or the Bernoulli and entropy functions, at the inlet for the smooth case. Then the existence for the smooth case can be applied to construct approximate solutions to establish the existence of weak solutions with vortex sheets/entropy waves by the compensated compactness argument. This is the first result on the global existence of solutions of the multidimensional steady compressible full Euler equations with free boundaries, which are not necessarily small perturbations of piecewise constant background solutions. The subsonic-sonic limit of the solutions is also shown. Finally, through the incompressible limit, the existence and uniqueness of incompressible Euler flows is established in arbitrary infinitely long nozzles for both the smooth solutions with large vorticity and the weak solution with vortex sheets. This is the joint work with Gui-Qiang G. Chen, Fei-Min Huang, and Wei Xiang. |
| Franziska | Weber | A sufficient condition for the Kolmogorov 4/5 law for stationary martingale solutions to the 3D Navier-Stokes equations | (joint work with: Jacob Bedrossian, Michele Coti Zelati, and Samuel Punshon-Smith) Abstract: We prove that statistically stationary martingale solutions of the 3D Navier-Stokes equations on the torus subjected to white-in-time (colored-in-space) forcing satisfy the Kolmogorov 4/5 law (in an averaged sense and over a suitable inertial range) using only the assumption that the kinetic energy is o(v^(-1)) as ν → 0 (where ν is the inverse Reynolds number). This plays the role of a weak anomalous dissipation. No energy balance or additional regularity is assumed (aside from that satisfied by all martingale solutions from the energy inequality). If the force is statistically homogeneous, then any homogeneous martingale solution satisfies the spherically averaged 4/5 law pointwise in space. An additional hypothesis of approximate isotropy in the inertial range gives the traditional version of the Kolmogorov law. We demonstrate a necessary condition by proving that energy balance and an additional quantitative regularity estimate as ν → 0 imply that the 4/5 law (or any similar scaling law) cannot hold. |
| Qinghua | Xiao | The Vlasov-Poisson-Boltzmann System for the Whole Range of Cutoff Soft Potentials | The dynamics of dilute electrons can be modeled by the fundamental one-species Vlasov-Poisson-Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electrostatic field. By introducing a new time-velocity weighted energy method and based on some new optimal temporal decay estimates on the solution itself and some of its derivatives with respect to both the spatial and the velocity variables, it is shown in this manuscript that the Cauchy problem of the one-species Vlasov-Poisson-Boltzmann system for all cutoff soft potentials does exist a unique global smooth solution for general initial perturbation which is unnecessary to satisfy the neutral condition but is assumed to be small in certain weighted Sobolev spaces. Our approach applies also to the case of cutoff hard potentials and thus provides a satisfactory global well-posedness theory to the one-species Vlasov-Poisson-Boltzmann system near Maxwellians for the whole range of cutoff intermolecular interactions in the perturbative framework. |
| Difan | Yuan | Global Entropy Solutions to Multi-Dimensional Isentropic Gas Dynamics with Spherical Symmetry | We are concerned with spherically symmetric solutions to the Euler equations for the multi-dimensional compressible fluids, which have many applications in diverse real physical situations. The system can be reduced to one dimensional isentropic gas dynamics with geometric source terms. Limited work has been done in the case including the origin, due to the presence of the singularity at the origin. In this talk, we present two existence theorems of global entropy solutions. The first one focuses on the case excluding the origin in which the negative velocity is allowed, and the second one is corresponding to the case including the origin with non-negative velocity. The compensated compactness framework and vanishing viscosity method are applied to prove the convergence of approximate solutions. In the second case, we show that if the blast wave initially moves outwards and the initial densities and velocities decay to zero with certain rate near origin, then the densities and velocities tend to zero with the same rate near the origin for any positive time. In particular, the entropy solutions in the two existence theorems are uniformly bounded with respect to time. |
| Dimitrios | Zacharenakis | A posteriori error estimates for a discontinuous Galerkin approximation of the Navier-Stokes-Korteweg model | In HYP2016 we presented an a posteriori error analysis of a discontinuous Galerkin scheme for the approximation of the isothermal Euler-Korteweg model in one space-dimension. In this talk, we extend the analysis to the one dimensional Navier-Stokes-Korteweg system, which is given by \begin{align} \begin{split} \rho_{t} + \left(\rho v\right)_{x} &= 0 \\ \left(\rho v \right)_{t} + \left(\rho v^{2} +p (\rho) \right)_{x} &= \mu v_{xx} + \gamma \rho \rho_{xxx} \end{split} \quad \text{in} \ S^{1} \times [0,T), \label{relativentropy} \tag{NSK} \end{align} with the mass density $\rho = \rho(x,t) >0$ and the velocity field $v = v(x,t)$ being the unknown quantities and $S^{1}$ denotes the unit interval with periodic boundary conditions. The viscosity coefficient and a final time are parameters denoted as $\mu \geq 0$ and $T >0$ respectively. We model capillarity effects by including density gradients in the energy, leading to the term $\gamma \rho \rho_{xxx}$ in (\ref{relativentropy}), where $\gamma > 0$ is a capillarity constant. Furthermore, the pressure, denoted by $p$, is a non-monotone function and is related to the Helmholtz free energy $p (\rho) = \rho W'(\rho) - W(\rho)$. We discretize (\ref{relativentropy}) using a local DG formulation in space and an explicit Runge-Kutta scheme in time. A possible approach to bound the difference between the exact and the numerical solution would be to use the relative entropy technique. However, the energy density $W$ has a multi-well shape. Thus, a fundamental component of our analysis is the reduced relative entropy stability framework, and to be able to apply it we introduce sufficiently regular intermediate functions called reconstructions (\cite{MN06}). Consequently, we can achieve our goal to estimate the difference between the numerical and the exact solutions. We present numerical tests of the error estimator's efficiency. In the future, we plan to employ mesh adaptive schemes based on this estimator. \end{Abstract} \begin{thebibliography}{99.} \bibitem{GP14b} J. Giesselmann, T. Pryer, \textit{Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics}, IMA J. Numer. Anal., vol. 36, no. 4, pp. 1685 - 1714, \textbf{2016} \par\vspace{2mm} \bibitem{MN06} C. Makridakis, R. H. Nochetto, \textit{A posteriori error analysis for higher order dissipative methods for evolution problems}, Numer. Math., vol. 104, no. 4, pp. 489 - 514, \textbf{2006} \par\vspace{2mm} \end{thebibliography} |
| Yanni | Zeng | Hyperbolic-parabolic balance laws with application | We consider a general system of hyperbolic-parabolic balance laws. Under a set of structural conditions, we are able to establish global existence of solution to the Cauchy problem around a constant equilibrium state. L^p (p >= 2) decay rates of the solution to the constant equilibrium state are obtained. In the case of multi space dimensions we may further discuss the asymptotic behavior of solution in L^p framework. The results apply to polyatomic gas flows with both translational and thermal non-equilibrium, and to a Keller-Segel chemotaxis model with logarithmic sensitivity and logistic growth. |
| Markus | Zenk | A Well-Balanced and low Diffusive Relaxation Scheme for atmospheric Applications | In this work we concern the approximation of solutions to the Euler equations with gravity in the low Mach number regime. To this end we use a finite volume discretization where for the numerical fluxes we use a modified Suliciu relaxation technique. In the describe regime we face two challanges. First, we are intereseted in computing small deviations from a hydrostatic equilibirum. Therefore we want to use a well-balanced method. Second, we seek for a low diffusive scheme especially in the low Mach number regime. We derive the upwind scheme and show its applicability by conducting various numerical experiments. |
| Qingtian | Zhang | ||
| Xiongtao | Zhang | Complete cluster predictability of the Cucker-Smale flocking model on the real line | We present a complete predictability of clustering for the Cucker-Smale(C-S) model on the line. Emergence of multi-cluster flocking is often observed in numerical simulations for the C-S model with short-range interactions. However, the explicit computation of the number of emergent multi-clusters a priori is a challenging problem. In this talk, I will present an explicit criterion and algorithm to calculate the number of clusters, their bulk velocities and critical value at which transition occurs in one-dimensional setting. I also provide several numerical examples and show the applications of this work. |
| Hao | Zheng | Global Existence, Uniqueness and Stability of Solutions to One-dimensonal Quantum Hydrodynamic Equations | Abstract In this paper we consider the Cauchy problem for the one-dimensional quantum hydrodynamic (QHD) system, namely the compressible Euler equations with a quantum correction term. We consider finite energy initial data with certain higher order bounds, which imply the square integrability of the chemical potential. These type of models have been extensively used to investigate Superfluidity, Superconductivity and recently in the modeling of senuconductor devices. The main result is the global existence and uniqueness of weak solutions to the Cauchy problem. Our approach does not require the initial data to be given through a wave function, as in previous results. Nevertheless the higher order bounds allow to construct a related wave function. Moreover we prove such higher order quantities are bounded for any finite time. Futhermore, those a priori estimates allow us to prove the stability of weak solutions to the QHD system. |
| David | Zorio Ventura | WENO reconstructions of unconditionally optimal order | A modified Weighted Essentially Non-Oscillatory (WENO) technique preventing accuracy loss near smooth extrema, regardless of their order, is presented. This approach uses only local data from around the reconstruction stencil. The resulting weights to account for discontinuities are non-dimensional and scale-independent. We provide two different ways to define the weights and prove that both give the sought accuracy order. Some numerical experiments are performed showing that the modified third order schemes outperform the classical third order WENO schemes and even higher order WENO schemes in terms of the ratio CPU cost/error. This is a joint work with Antonio Baeza (Universitat de Val?ncia, Spain), Raimund B?rger (Universidad de Concepci?n, Chile) and Pep Mulet (Universitat de Val?ncia, Spain). |
| Philipp | Öffner | Error boundedness of Correction Procedure via Reconstruction / Flux Reconstruction and the Connection to Residual Distribution Schemes | In this talk, we focus on the correction procedure via reconstruction (CPR) / flux reconstruction (FR) methods for hyperbolic conservation laws and investigate properties of these methods. We start by studying the long time error behavior of the CPR/ FR methods and derive an error equation for a model problem. We investigate this error equation in detail. We demonstrate that not only the choice of the numerical flux (upwind or central)affects the growth rate and asymptotic value of the error, but also that the selection of bases (Gauss-Lobatto or Gauss-Legendre) highly impacts the solution. Using Gauss-Legendre basis, the error reaches the asymptotic value faster and a lower value than by using Gau\ss-Lobatto basis. In the second part of this talk, we demonstrate that flux reconstruction can be transformed into the Resdiual Distribuation (RD) framework and vice versa. Because of this closed connection we are able to apply known results from RD schemes to CPR/ FR methods. In this context, we focus on entropy stability. The first part of this talk is joined work with H. Ranocha from Technical University of Brunswick (Germany) whereas the second part is in cooperation with R. Abgrall and E. le M?l?do from the University of Z?rich (Switzerland). |
Acceptance of contributed talks is at the discretion of the Scientific Committee.
April 15, 2018 is the deadline for abstract submission of all contributed talks (deadline has been extended).
Image of “Old Main”, courtesy of Penn State Board of Trustees.