Jean-Marc Hérard

Some approximate Godunov schemes to compute shallow-water equations with topography

Abstract We study here the computation of shallow-water equations with topography by Finite Volume methods, in a one-dimensional framework (though all methods introduced may be naturally extended in two dimensions). All methods are based on a discretisation of the topography by a piecewise function constant on each cell of the mesh, from an original idea of Le Roux et al. Whereas the Well-Balanced scheme of Le Roux is based on the exact resolution of each Riemann problem, we consider here approximate Riemann solvers. Several single step methods are derived from this formalism, and numerical results are compared to a fractional step method. Some test cases are presented: convergence towards steady states in subcritical and supercritical configurations, occurrence of dry area by a drain over a bump and occurrence of vacuum by a double rarefaction wave over a step. Numerical schemes, combined with an appropriate high-order extension, provide accurate and convergent approximations.

Closure laws for a two-fluid two-pressure model

Closure laws for interfacial pressure and interfacial velocity are proposed within the frame work of two-pressure two-phase flow models. These enable us to ensure positivity of void fractions, mass fractions and internal energies when investigating field by field waves in the Riemann problem. To cite this article: F. Coquel et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 927–932.

A Naive Riemann Solver to Compute a Non-conservative Hyperbolic System

An approximate Riemann solver is presented in this contribution, which enables to compute shock waves in compressible flows using one or two-equation turbulence model.

Comparaison de solveurs numériques pour simuler la turbulence compressible

We focus on the computation of the hyperbolic system describing a turbulent flow for isentropic gas. An exact Riemann solver is computed, we construct the solution of the Riemann problem and thus we derive a Godunov scheme. We provide some numerical simulations to exhibit a comparison between Godunov scheme and two approximate solvers: Vfroe and Rusanov scheme.© 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS

Part II. Comparison of Numerical Solvers for a Multicomponent Turbulent Flow

We focus on the computation of a hyperbolic system describing a multicomponent turbulent flow for isentropic gases, using an exact Riemann solver. This method is very robust, but costly. Thus, we introduce two approximate upwinding schemes: a Godunov scheme called VFRoe and a Rusanov scheme. The Rusanov scheme always ensures positive values for mass, concentration and turbulent kinetic energy, but generates less accurate results. We show some one- and two-dimensional computations and compare these three resolution methods.

Hybrid schemes to compute contact discontinuities in Euler equations with any EOS

We propose here some explicit hybrid schemes which enable accurate computation of Euler equations with arbitrary (analytic or tabulated) equation of state (EOS). The method is valid for the exact Godunov scheme and some approximate Godunov schemes.