Multidimensional HLLE Riemann solver; Application to Euler and Magnetohydrodynamic Flows

摘要

In this work we present a general strategy for constructing multidimensionalRiemann solvers with a single intermediate state, with particular attentionpaid to detailing the two-dimensional Riemann solver. This is accomplished byintroducing 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 facilitatenumerical implementation. The Riemann solver is proved to be positivelyconservative for the density variable; the positivity of the pressure variablehas been demonstrated for Euler flows when the divergence in the fluidvelocities is suitably restricted so as to prevent the formation of cavitationin the flow. We also focus on the construction of multidimensionally upwinded electricfields for divergence-free magnetohydrodynamical flows. A robust and efficientsecond order accurate numerical scheme for two and three dimensional Euler andmagnetohydrodynamic flows is presented. The scheme is built on the currentmultidimensional Riemann solver. The number of zones updated per second by thisscheme on a modern processor is shown to be cost competitive with schemes thatare based on a one-dimensional Riemann solver. However, the present schemepermits larger timesteps.