Nondiffusive conservative schemes based on approximate Riemann solvers for Lagrangian gas dynamics

¹ Sorbonne Universités, UPMC University Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place 75005 Jussieu, Paris, France.
nina.aguillon@upmc.fr
² Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France.
christophe.chalons@uvsq.fr

Received: 10 November 2014
Revised: 23 October 2015
Accepted: 25 January 2016

Abstract

In this paper, we present a conservative finite volume scheme for the gas dynamics in Lagrangian coordinates, which is fast and nondiffusive. By fast, we mean that it relies on an approximate Riemann solver, and hence the costly resolution of Riemann problems is avoided. By nondiffusive, we mean that the solution provided by the scheme is exact when the initial data is an isolated admissible shock, and discontinuities are sharply captured in general. The construction of the scheme uses two main tools: the approximate Riemann solver of [Ch. Chalons and F. Coquel, Math. Models Methods Appl. Sci. 24 (2014) 937–971.], which turns out to be exact on isolated admissible shocks, and a discontinuous reconstruction strategy, which consists in rebuilding entropy satisfying shocks inside some well chosen cells. Numerical experiments in 1D and 2D are proposed.