A simple two-dimensional extension of the HLL Riemann solver for hyperbolic systems of conservation laws

Abstract

We derive a simple method to numerically approximate the solution of the two-dimensional Riemann problem for gas dynamics, using the literal extension of the well-known HLL formalism as its basis. Essentially, any strategy attempting to extend the three-state HLL Riemann solver to multiple space dimensions will by some means involve a piecewise constant approximation of the complex two-dimensional interaction of waves, and our numerical scheme is not the exception. In order to determine closed form expressions for the involved fluxes, we rely on the equivalence between the consistency condition and the use of Rankine–Hugoniot conditions that hold across the outermost waves. The proposed scheme is carefully designed to simplify its eventual numerical implementation and its advantages are analytically attested. In addition, we show that the proposed solver can be applied to obtain the edge-centered electric fields needed in the constrained transport technique for the ideal magnetohydrodynamic (MHD) equations. We present several numerical results for hydrodynamics and magnetohydrodynamics that display the scheme's accuracy and its ability to be applied to various systems of conservation laws.