An accurate and efficient Riemann solver with tangent velocities for Godunov schemes in special relativistic hydrodynamics

In order to numerically integrate the special relativistic hydrodynamics (SRHD) equations through a Godunov scheme, the use of an exact Riemann solver having a low computational cost is proposed. The solver works through a Newton iterative method in pressure, by using a preliminary detection of the wave pattern and an integration by series of the ODE for the normal velocity across a rarefaction wave. A comparison with the solver of Marti and Muller shows about a 90% reduction of the computational cost. The present Riemann solver has been used inside a first-order Godunov scheme, in order to test the solver for high jumps of the primitive variables, perhaps avoiding certain numerical manipulations (e.g. limiting filters and multi-step integration schemes). The time evolution of a two-dimensional mixing layer is finally considered and a comparison with the solution obtained from a HLL method is carried out.The main result of this analysis lies in showing how the present exact Riemann solver can be successfully employed in the effective numerical integration of the SRHD equations