Geodesic
Nondiffusive conservative schemes based on approximate Riemann solvers for Lagrangian gas dynamics
Aguillon, Nina1  ; Chalons, Christophe2 
1 Sorbonne Universités, UPMC University Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place 75005 Jussieu, Paris, France.
2 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1887-1916

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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.

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DOI : 10.1051/m2an/2016010
Classification : 35L65, 35L40, 65M08, 76N15, 76M12
Keywords: Conservative finite volume scheme, discontinuous reconstruction, approximate Riemann solver, non diffusive scheme, Sharp discontinuities

Aguillon, Nina 1 ; Chalons, Christophe 2

1 Sorbonne Universités, UPMC University Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4 place 75005 Jussieu, Paris, France.
2 Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 78035 Versailles, France. 
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     title = {Nondiffusive conservative schemes based on approximate {Riemann} solvers for {Lagrangian} gas dynamics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1887--1916},
     publisher = {EDP-Sciences},
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Aguillon, Nina; Chalons, Christophe. Nondiffusive conservative schemes based on approximate Riemann solvers for Lagrangian gas dynamics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1887-1916. doi: 10.1051/m2an/2016010

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