Multidimensional HLLE Riemann solver; Application to Euler and Magnetohydrodynamic Flows

TL;DR

This paper introduces a multidimensional Riemann solver with a single intermediate state for Euler and MHD flows, providing a robust, efficient scheme that allows larger timesteps and maintains positivity and divergence-free conditions.

Contribution

It develops a novel multidimensional Riemann solver with closed-form flux expressions and demonstrates its effectiveness for Euler and MHD simulations.

Findings

Positively conservative for density

Maintains pressure positivity under certain conditions

Cost-competitive with 1D Riemann-based schemes, with larger timesteps

Abstract

In this work we present a general strategy for constructing multidimensional Riemann solvers with a single intermediate state, with particular attention paid to detailing 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 also provided to facilitate numerical implementation. The Riemann solver is proved to be positively conservative for the density variable; the positivity of the pressure variable has been demonstrated for Euler flows when the divergence in the fluid velocities is suitably restricted so as to prevent the formation of cavitation in the flow. We also focus on the construction of multidimensionally upwinded electric fields for divergence-free magnetohydrodynamical flows. A robust and efficient second order accurate numerical scheme for two and three dimensional Euler and magnetohydrodynamic flows is presented. The scheme is built on the current multidimensional Riemann solver. 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.