Multidimensional HLLC Riemann solver for unstructured meshes – with application to Euler and MHD flows

Abstract

The goal of this paper is to formulate genuinely multidimensional HLL and HLLC Riemann solvers for unstructured meshes by extending our prior papers on the same topic for logically rectangular meshes Balsara (2010, 2012) [4,5]. Such Riemann solvers operate at each vertex of a mesh and accept as an input the set of states that come together at that vertex. The mesh geometry around that vertex is also one of the inputs of the Riemann solver. The outputs are the resolved state and multidimensionally upwinded fluxes in both directions. A formulation which respects the detailed geometry of the unstructured mesh is presented. Closed-form expressions are provided for all the integrals, making it particularly easy to implement the present multidimensional Riemann solvers in existing numerical codes. While it is visually demonstrated for three states coming together at a vertex, our formulation is general enough to treat multiple states (or zones with arbitrary geometry) coming together at a vertex. The present formulation is very useful for two-dimensional and three-dimensional unstructured mesh calculations of conservation laws. It has been demonstrated to work with second to fourth order finite volume schemes on two-dimensional unstructured meshes. On general triangular grids an arbitrary number of states might come together at a vertex of the primal mesh, while for calculations on the dual mesh usually three states come together at a grid vertex. We apply the multidimensional Riemann solvers to hydrodynamics and magnetohydrodynamics (MHD) on unstructured meshes. The Riemann solver is shown to operate well for traditional second order accurate total variation diminishing (TVD) schemes as well as for weighted essentially non-oscillatory (WENO) schemes with ADER (Arbitrary DERivatives in space and time) time-stepping. Several stringent applications for compressible gasdynamics and magnetohydrodynamics are presented, showing that the method performs very well and reaches high order of accuracy in both space and time. The present multidimensional Riemann solver is cost-competitive with traditional, one-dimensional Riemann solvers. It offers the twin advantages of isotropic propagation of flow features and a larger CFL number.