Nicolas Seguin

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.

ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

NUMERICAL MODELING OF TWO-PHASE FLOWS USING THE TWO-FLUID TWO-PRESSURE APPROACH

The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows.

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.

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 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.

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.

Error Estimate for Godunov Approximation of Locally Constrained Conservation Laws

We consider a model of traffic flow with unilateral constraint on the flux introduced by Colombo and Goatin [J. Differential Equations, 234 (2007), pp. 654--675], for which the convergence of numerical approximation using monotone finite volume schemes has been performed by Andreianov, Goatin, and Seguin [Numer. Math., 115 (2010), pp. 609--645]. We derive for this problem a new

Weak solutions to Friedrichs systems with convex constraints

{\rm BV}