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A two-dimensional HLLC Riemann solver for conservation laws: application to Euler and magnetohydrodynamic flows. (English) Zbl 1284.76261
Summary: In this paper we present a genuinely two-dimensional HLLC Riemann solver. On logically rectangular meshes, it accepts four input states that come together at an edge and outputs the multi-dimensionally upwinded fluxes in both directions. This work builds on, and improves, our prior work on two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here achieves its stabilization by introducing a constant state in the region of strong interaction, where four one-dimensional Riemann problems interact vigorously with one another. A robust version of the HLL Riemann solver is presented here along with a strategy for introducing sub-structure in the strongly-interacting state. Introducing sub-structure turns the two-dimensional HLL Riemann solver into a two-dimensional HLLC Riemann solver. The sub-structure that we introduce represents a contact discontinuity which can be oriented in any direction relative to the mesh.
The Riemann solver presented here is general and can work with any system of conservation laws. We also present a second order accurate Godunov scheme that works in three dimensions and is entirely based on the present multidimensional HLLC Riemann solver technology. The methods presented are cost-competitive with traditional higher order Godunov schemes.
The two-dimensional HLLC Riemann solver is shown to work robustly for Euler and Magnetohydrodynamic (MHD) flows. Several stringent test problems are presented to show that the inclusion of genuinely multidimensional effects into higher order Godunov schemes indeed produces some very compelling advantages. For two dimensional problems, we were routinely able to run simulations with CFL numbers of $$\sim$$0.7, with some two-dimensional simulations capable of reaching higher CFL numbers. For three dimensional problems, CFL numbers as high as $$\sim$$0.6 were found to be stable. We show that on resolution-starved meshes, the scheme presented here outperforms unsplit second order Godunov schemes that are based on conventional one-dimensional Riemann solver technology. Strong discontinuities are shown to propagate very isotropically using the methods presented here. The present Riemann solver provides an elegant resolution to the problem of obtaining multi-dimensionally upwinded electric fields in MHD without resorting to a doubling of the dissipation in each dimension.

##### MSC:
 76M12 Finite volume methods applied to problems in fluid mechanics 35L65 Hyperbolic conservation laws 76W05 Magnetohydrodynamics and electrohydrodynamics 35Q31 Euler equations
RIEMANN; HLLE
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##### References:
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