A high-order multi-dimensional HLL-Riemann solver for non-linear Euler equations

Abstract

This article presents a numerical model that enables to solve on unstructured triangular meshes and with a high-order of accuracy, a multi-dimensional Riemann problem that appears when solving hyperbolic problems. For this purpose, we use a MUSCL-like procedure in a “cell-vertex” finite-volume framework. In the first part of this procedure, we devise a four-state bi-dimensional HLL solver (HLL-2D). This solver is based upon the Riemann problem generated at the centre of gravity of a triangular cell, from surrounding cell-averages. A new three-wave model makes it possible to solve this problem, approximately. A first-order version of the bi-dimensional Riemann solver is then generated for discretizing the full compressible Euler equations. In the second part of the MUSCL procedure, we develop a polynomial reconstruction that uses all the surrounding numerical data of a given point, to give at best third-order accuracy. The resulting over determined system is solved by using a least-square methodology. To enforce monotonicity conditions into the polynomial interpolation, we develop a simplified central WENO (CWENO) procedure. Numerical tests and comparisons with competing numerical methods enable to identify the salient features of the whole model.