An entropy satisfying MUSCL scheme for systems of conservation laws

Abstract

For the high-order numerical approximation of hyperbolic systems of conservation laws, we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations. Based on this new criterion, we derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities. The resulting scheme is a nonlinear version of the classical Van Leer's MUSCL scheme. Strong convergence of this second order, entropy satisfying scheme is proved for systems of two equations. Numerical tests demonstrate the interest of our theory.

Second‐order entropy diminishing scheme for the Euler equations

“…on each cell C i , we can suppose, for simplicity, that C i =]0, 1[. The projection problem can then be presented in the following abstract setting (introduced in [13], [7]):…”

“…It is based on the following notion of (what we decided to call) pseudo-derivative which was introduced first by Bouchut, Bourdarias and Perthame in [13] in the case of a scalar equation. It was also studied by Coquel and Le Floch in [7] for systems.…”

“…An important feature is that this stability criterion will be verified for the whole family of entropies of the system. Such an approach has been initiated for scalar conservation laws by Bouchut, Bourdarias and Perthame in [13] and has been extended to 2x2 systems of conservation laws and to the 3x3 system of Lagrangian gas dynamics by Coquel and Le Floch in [7]. Our paper deals with the case of the Euler system.…”

“…One approach could have been to develop a generalized Riemann solver as in [7] for the gas dynamics equations in Lagrangian coordinates. The theory of the generalized Riemann problem can be found, for example, in the papers of Bourgeade, Le Floch, Raviart [5], Ben Artzi, Falcovitz [2].…”

“…In several papers of Bouchut, Bourdarias, Perthame and Coquel, Le Floch ([13], [7]...), a general methodology has been developed to construct second order finite volume schemes for hyperbolic systems of conservation laws satisfying the entire family of entropy inequalities. This approach is mainly based on the construction of an entropy diminishing projection.…”