A Simple Two-Dimensional Extension of the HLL Riemann Solver for Gas Dynamics

We report on our study aimed at deriving 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 planar waves emerging from the Riemann problem's initial discontinuities. The proposed scheme is then carefully designed to simplify its eventual numerical implementation and its advantages are analytically attested. We also present first numerical results that put into evidence its robustness and stability.