An essential ingredient of the HLL scheme are good estimates for the smallest and largest signal velocities. In the non-relativistic case, Einfeldt  proposed calculating them based on the smallest and largest eigenvalues of Roe’s matrix. The HLL scheme with Einfeldt’s recipe (HLLE) is a very robust upwind scheme for the Euler equations and possesses the property of being positively conservative. The HLLE method is exact for single shocks, but it is very dissipative, especially at contact discontinuities.
Schneider et al.  have presented results in 1D relativistic hydrodynamics using a relativistic version of the HLL method (RHLLE) with signal velocities given bycs is the relativistic sound speed, and where the bar denotes the arithmetic mean between the initial left and right states. Duncan and Hughes  have generalized this method to 2D SRHD and applied it to the simulation of relativistic extragalactic jets.
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