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

Summary This paper describes a numerical discretization of the compressible Euler equations with gravitational potential. A pertinent feature solutions to these inhomogeneous is special case stationary zero velocity, described by nonlinear partial differential equation, whose are called hydrostatic equilibria. We present well‐balanced method, meaning that besides discretizing complete equations, method also able maintain all The finite volume Riemann solver approximated so‐called relaxation...

The VFRoe scheme has been recently introduced by Buffard, Gallouët, and Hérard [Comput. Fluids, 29 (2000), pp. 813–847] to approximate the solutions of shallow water equations. One main interests this method is be easily implemented. As a consequence, such appears as an interesting alternative other more sophisticated schemes. methods perform in good agreement with expected ones. However, robustness numerical procedure not proposed. Following ideas Jin Xin [Comm. Pure Appl. Math., 45 (1995),...

This work concerns the derivation of HLL schemes to approximate solutions systems conservation laws supplemented by source terms. Such a system contains many models such as Euler equations with high friction or M1 model for radiative transfer. The main difficulty arising from these comes particular asymptotic behavior. Indeed, in limit some suitable parameter, tends diffusion equation. article is devoted derive methods able associated transport regime but also restore diffusive regime. To...

The present paper concerns the derivation of numerical schemes to approximate weak solutions Ripa model, which is an extension shallow-water model where a gradient temperature considered. Here, main motivation lies in exact capture steady states involved model. Because gradient, at rest, prime importance from physical point view, turn out be very nonlinear and their by scheme challenging. We propose relaxation technique derive required scheme. In fact, we exhibit Riemann solver that...

This work concerns the design of well-balanced entropy stable numerical schemes for shallow water equations. The fully discrete inequality is reached by introducing a local condition incorporated in scheme design. source term discretized to preserve both steady states and stability. method yields explicit which are relevantly illustrated with several test cases.

The present work concerns the derivation of a numerical scheme to approximate weak solutions Euler equations with gravitational source term. designed is proved be fully well-balanced since it able exactly preserve all moving equilibrium solutions, as well corresponding steady at rest obtained when velocity vanishes. Moreover, proposed entropy-stable satisfies discrete entropy inequalities. In addition, in order satisfy required admissibility positivity both density and pressure established....

For a class of non-conservative hyperbolic systems partial differential equations endowed with strictly convex mathematical entropy, we formulate the initial-value problem by supplementing kinetic relation prescribing rate entropy dissipation across shock waves. Our condition can be regarded as generalization to similar concept introduced Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions LeFloch non-classical undercompressive shocks nonlinear systems. The proposed turns out...

Abstract In the present work, we consider numerical approximation of pressureless gas dynamics in one and two spatial dimensions. Two particular phenomena are special interest for us, namely δ‐shocks vacuum states. A relaxation scheme is developed which reliably captures these phenomena. space dimension, prove validity several stability criteria, i.e., show that a maximum principle as well TVD property discrete velocity component entropy inequalities hold. Several tests considering not only...

We propose a numerical scheme to approximate the weak solutions of 10-moment Gaussian closure. The moment closure for gas dynamics is governed by conservative hyperbolic system supplemented entropy inequalities whose satisfy positiveness density and tensorial pressure. consider Suliciu-type relaxation solutions. These methods are proved all expected properties discrete inequalities. illustrated several experiments.

We investigate the late-time asymptotic behavior of solutions to nonlinear hyperbolic systems conservation laws containing stiff relaxation terms. First, we introduce a Chapman-Enskog-type expansion and derive an effective system equations describing late-time/stiff-relaxation singular limit. The structure this new is discussed role mathematical entropy emphasized. Second, propose finite volume discretization which, in asymptotics, allows us recover discrete version same system. This...

In this note we are interested in the modelling of sediment transport phenomena. We mostly focus on bedload and do not consider suspension processes. first propose a numerical scheme for classical Saint-Venant – Exner model. It is based relaxation approach whole system it works with all flux function. The stability investigated some tests proposed. exhibit that coupled more stable than splitting used industrial softwares. Then derive an original three layers model order to overcome...