A third-order entropy stable scheme for hyperbolic conservation laws (Q2798339)

The authors consider a system of nonlinear conservation laws in \(d\) dimensions along with an entropy condition (assuming the existence of an entropy pair \(\eta, q\) satisfying the usual inequality) and investigate a semi-discrete difference method. For the numerical flux, they base on the classical paper of \textit{E. Tadmor} [Math. Comput. 49, 91--103 (1987; Zbl 0641.65068)], and turn to higher-order (spec. third-order) approximate solutions following \textit{P. G. LeFloch} et al. [SIAM J. Numer. Anal. 40, No. 5, 1968--1992 (2002; Zbl 1033.65073)], by linearly combining two-point entropy conservative fluxes, adding a dissipation term and using a nonlinear limiter. The dissipation term is obtained by piecewise third-order reconstruction using point values.NEWLINENEWLINEThe authors show that their semi-discrete method is entropy stable. By working dimensionwise they generalize their approach to many dimensions. They also report on numerical experiments in which the time discretization is done by SSP Runge-Kutta. In 1D (examples include Sod, Burgers, Toro) overshoots are removed by modifying the limiter. In 2D they consider the Euler equations and follow the advection of a vortex, moreover, two Riemann problems are solved.NEWLINENEWLINELet us mention that no hint is given that the entropy condition may be insufficient if \(d>1\), for this see, e.g. [\textit{U. S. Fjordholm} et al., ``Construction of approximate entropy measure valued solutions for hyperbolic systems of conservation laws'', Preprint, \url{arXiv:1402.0909}] or [\textit{V. Elling}, in: Hyperbolic problems. Theory, numerics and applications. Vol. 1. Proceedings of the 13th international conference on hyperbolic problems, HYP 2010, Beijing, China, June 15--19, 2010. Hackensack, NJ: World Scientific; Beijing: Higher Education Press. Series in Contemporary Applied Mathematics CAM 17, 203--214 (2012; Zbl 1284.35308)].