Journal
JOURNAL OF COMPUTATIONAL PHYSICS
Volume 229, Issue 6, Pages 1970-1993Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jcp.2009.11.018
Keywords
Riemann solvers; Multidimensional; Conservation laws; Euler; MHD
Funding
- NSF [AST-0607731, NSF-AST-0947765]
- NASA [NASA-NNX07AG93G, NASA-NNX08AG69G]
- Division Of Astronomical Sciences
- Direct For Mathematical & Physical Scien [1009091, 0947765] Funding Source: National Science Foundation
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In this work we present a general strategy for constructing multidimensional HLLE Riemann solvers, with particular attention paid to detailing the two-dimensional HLLE 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 (MHD) flows. A robust and efficient second order accurate numerical scheme for two and three-dimensional Euler and MHD flows is presented. The scheme is built on the current multidimensional Riemann solver and has been implemented in the author's RIEMANN code. 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. Accuracy analysis for multidimensional 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 per-form robustly on all of them. (C) 2009 Elsevier Inc. All rights reserved.
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