Three dimensional HLL Riemann solver for conservation laws on structured meshes; Application to Euler and magnetohydrodynamic flows

Abstract

In this paper we build on our prior work on multidimensional Riemann solvers by detailing the construction of a three-dimensional HLL Riemann solver. As with the two-dimensional Riemann solver, this is accomplished by introducing a constant resolved state between the states being considered, which introduces sufficient dissipation for systems of conservation laws. Closed form expressions for the resolved fluxes are provided to facilitate numerical implementation. This is accomplished by introducing a novel derivation of the multidimensional Riemann solver. The novelty consists of integrating Lagrangian fluxes across moving surfaces. This makes the problem easier to visualize in three dimensions. (A video introduction to multidimensional Riemann solvers is available on http://www.nd.edu/~dbalsara/Numerical-PDE-Course.)A robust and efficient second order accurate numerical scheme for three dimensional Euler and MHD flows is presented. The scheme is built on the current three-dimensional Riemann solver and has been implemented in the author's RIEMANN code. We demonstrate that schemes that are based on the three-dimensional Riemann solver permit multidimensional discontinuities to propagate more isotropically on resolution-starved meshes. The number of zones updated per second by this scheme on a modern processor is shown to be cost competitive with schemes that are based on a one-dimensional Riemann solver. However, the present scheme permits larger timesteps in three dimensions because of its inclusion of genuinely three-dimensional effects in the flow. For MHD problems it is not necessary to double the dissipation when evaluating the edge-centered electric fields.Accuracy analysis for three-dimensional Euler and MHD problems shows that the scheme meets its design accuracy. Several stringent test problems involving Euler and MHD flows are also presented and the scheme is shown to perform robustly on all of them. For the very first time, we present the formulation and solution of three-dimensional Riemann problems.