Edwige Godlewski

The numerical interface coupling of nonlinear hyperbolic systems of conservation laws: I. The scalar case

SummaryWe study the theoretical and numerical coupling of two general hyperbolic conservation laws. The coupling preserves in a weak sense the continuity of the solution at the interface without imposing the overall conservativity of the coupled model. In order to analyze the convergence of the coupled numerical scheme, we first revisit the approximation of the boundary value problems. We then prove the convergence and characterize the limit solution of the coupled schemes in a few simple but significative coupling situations. The general coupling problem is analyzed for Riemann initial data and illustrated by numerical simulations. Résumé. Nous nous intéressons à une nouvelle forme de couplage de deux systèmes hyperboliques de lois de conservation. Ce couplage assure de façon faible la continuité de la solution à l’interface sans imposer la conservativité du modèle couplé. Pour étudier la convergence du schéma d’approximation numérique, nous commençons par reprendre les résultats concernant l’approximation du problème aux limites. Nous démontrons ensuite la convergence du schéma couplé dans un certain nombre de cas intéressants. Le cas général du couplage est étudié et illustré numériquement pour une donnée initiale de Riemann.

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.

The linearized stability of solutions of nonlinear hyperbolic systems of conservation laws A general numerical approach

We study the linearized stability of a discontinuous solution of a multidimensional hyperbolic system of conservation laws by linearizing the system around the basic solution; the resulting linearized system has discontinuous coefficients and involves nonconservative products. We propose a direct approach of the problem which introduces measure solutions and gives a natural meaning to the nonconservative product. This approach leads to simple numerical schemes.

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.

RELAXATION OF FLUID SYSTEMS

We propose a relaxation framework for general fluid models which can be understood as a natural extension of the Suliciu approach in the Euler setting. In particular, the relaxation system may be totally degenerate. Several stability properties are proved. The relaxation procedure is shown to be efficient in the numerical approximation of the entropy weak solutions of the original PDEs. The numerical method is particularly simple in the case of a fully degenerate relaxation system for which the solution of the Riemann problem is explicit. Indeed, the Godunov solver for the homogeneous relaxation system results in an HLLC-type solver for the equilibrium model. Discrete entropy inequalities are established under a natural Gibbs principle.

On the linearization of systems of conservation laws for fluids at a material contact discontinuity

Abstract In order to investigate the linearized stability or instability of compressible flows, as it occurs for instance in Rayleigh–Taylor or Kelvin–Helmholtz instabilities, we consider the linearization at a material discontinuity of a flow modeled by a multidimensional nonlinear hyperbolic system of conservation laws. Restricting ourselves to the plane-symmetric case, the basic solution is thus a one-dimensional contact discontinuity and the normal modes of pertubations are solutions of the resulting linearized hyperbolic system with discontinuous nonconstant coefficients and source terms. While in Eulerian coordinates, the linearized Cauchy problem has no solution in the class of functions, we prove that for a large class of systems of conservation laws written in Lagrangian coordinates and including the Euler and the ideal M.H.D. systems, there exists a unique function solution of the problem that we construct by the method of characteristics.

A Relaxation Method for the Coupling of Systems of Conservation Laws

The present work investigates the spatial coupling of different hyperbolic models, from the theoritical and numerical points of view. The problem of coupling often occurs in an industrial context. Consider a complex system, such as a nuclear reactor. Different phenomena can appear according to the region and thus, different models (and numerical methods) are used. The difficulty lies in letting the different models communicate, since they are not necessarily compatible. Hence we study the spatial coupling of two different hyperbolic systems of conservation laws through the interface {x = 0}:

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

In numerous industrial CFD applications, it is usual to use two (or more) different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman–Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.

The case of multidimensional systems

We shall mainly consider in this section a two-dimensional p × p hyperbolic system (1.1) .

Asymptotic Preserving Scheme for Euler System with Large Friction

We construct a ‘well-balanced’ and ‘asymptotic preserving’ scheme for the approximation of the model problem of gas dynamics equations with gravity and friction. The friction terms we consider are quite general. We interpret our simple Riemann solver in such a way that the expected properties are directly inherited from the properties of the system of PDEs which is approximated.