Christophe Chalons

Godunov-type schemes for hyperbolic systems with parameter dependent source. The case of Euler system with friction

Well balanced or asymptotic preserving schemes are receiving an increasing amount of interest. This paper gives a precise setting for studying both properties in the case of Euler system with friction. We derive a simple solver which, by construction, preserves discrete equilibria and reproduces at the discrete level the same asymptotic behavior as that of the solutions of the continuous system. Numerical illustrations are convincing and show that not all methods share these properties.

Navier-Stokes equations with several independent pressure laws and explicit predictor-corrector schemes

This work is concerned with the numerical capture of stiff viscous shock solutions of Navier-Stokes equations for complex compressible materials, in the regime of large Reynolds numbers. After [2] and [6], a relevant numerical capture is known to require the satisfaction of an extended set of non classical Rankine-Hugoniot conditions due to the non conservation form of the governing PDE model. Here, we show how to enforce their validity at the discrete level without the need for solving local non linear algebraic problems. Non linearities are bypassed when introducing new averaging techniques which are proved to satisfy all the desirable stability properties when invoking suitable approximate Riemann solutions. A relaxation procedure is proposed to that purpose with the benefit of a fairly simple overall numerical method.

NUMERICAL APPROXIMATION OF A MACROSCOPIC MODEL OF PEDESTRIAN FLOWS

This paper is concerned with the numerical approximation of the solutions of a macroscopic model for the description of the flow of pedestrians. Solutions of the associated Riemann problem are known to be possibly nonclassical in the sense that the underlying discontinuities may well violate the Lax inequalities, which makes their numerical approximation very sensitive. This study proposes to extend the transport‐equilibrium strategy proposed in [C. Chalons, C. R. Acad. Sci. Paris Ser. I, 342 (2006), pp. 623–626] and [C. Chalons, Transport‐equilibrium schemes for computing nonclassical shocks. I. Scalar conservation laws, submitted] for computing the nonclassical solutions of scalar conservation laws with either a concave‐convex or a convex‐concave flux function and supplemented with an invertible kinetic function. These strong properties are not fulfilled in the present setting, since the flow function admits several inflection points and the kinetic function is not invertible. We nevertheless succeed in...

Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I

We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions.

Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling

A new version of Godunov scheme is proposed in order to compute solutions of a traffic flow model with phase transitions. The scheme is based on a modified averaging strategy and a sampling procedure. Several numerical tests are shown to prove the validity of the method. The convergence of the algorithm is demonstrated numerically. We also give a higher order extension of the method in space and time.

Computing undercompressive waves with the random choice scheme. Nonclassical shock waves

For several nonlinear hyperbolic models of interest we investigate the stability and largetime behavior of undercompressive shock waves characterized by a kinetic relation. The latter are considered as interfaces between two materials with distinct constitutive relations. We study nonclassical entropy solutions to scalar conservation laws with concave-convex flux-function and a non-genuinely nonlinear, strictly hyperbolic model of two conservation laws arising in nonlinear elastodynamics. We use Glimm’s random choice scheme but we replace the classical Riemann solver with the nonclassical one described recently in [21, 24]. Our numerical experiments demonstrate the robustness and accuracy of the random choice scheme for computing nonclassical shock waves which are known to be very sensitive to dissipation and dispersion mechanisms. In this paper, we study carefully various issues related to nonclassical shocks and their stability under perturbations. This numerical study yields important hints for further theoretical investigation on, for instance, the double N-wave pattern put forward when studying the time-asymptotic behavior of periodic nonclassical solutions.

The interface coupling of the gas dynamics equations

We investigate the one-dimensional coupling of two systems of gas dynamics at a fixed interface. The coupling constraints consist in requiring the continuity of a system of nonconservative variables at the interface. Since we are dealing with hyperbolic systems, weak coupling conditions are proposed. The existence and the uniqueness of the solutions of the coupled Riemann problem are investigated. Several examples of solutions satisfying the weak coupling conditions are contructed, either continuous or discontinuous with respect to the nonconservative variables at the interface.

Coupling of general Lagrangian systems

This work is devoted to the coupling of two fluid models, such a s two Euler systems in Lagrangian coordinates, at a fixed interface. We define coupling conditi ons which can be expressed in terms of continuity of some well chosen variables and then solve the coupled Riemann problem. In the present setting where the interface is characteristic, a particula r choice of dependent variables which are transmitted ensures the overall conservativity.

Fast Relaxation Solvers for Hyperbolic-Elliptic Phase Transition Problems

Phase transition problems in compressible media can be modelled by mixed hyperbolic-elliptic systems of conservation laws. Within this approach phase boundaries are understood as shock waves that satisfy additional constraints, sometimes called kinetic relations. For numerical approximation tracking-type algorithms have been suggested. The core piece of these algorithms is the usage of exact Riemann solvers incorporating the kinetic relation. However, exact Riemann solvers are computationally expensive or even not available. In this paper we present a class of approximate Riemann solvers for hyperbolic-elliptic models that relies on a generalized relaxation procedure. It preserves in particular the kinetic relation for phase boundaries exactly and gives for isolated phase transitions the correct solutions. In combination with a novel subiteration procedure the approximate Riemann solvers are used in the tracking algorithms. The efficiency of the approach is validated on a barotropic system with linear kin...

A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations

This work is devoted to the derivation of a fully well-balanced numerical scheme for the well-known shallow-water model. During the last two decades, several well-balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so-called lake at rest. By fully well-balanced, we mean here that the proposed Godunov-type method is also able to preserve stationary states with non zero velocity. The numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality.