where and are lower and upper bounds for the smallest and largest signal velocities, respectively. The intermediate state is determined by requiring consistency of the approximate Riemann solution with the integral form of the conservation laws in a grid zone. The resulting integral average of the Riemann solution between the slowest and fastest signals at some time is given by

and the numerical flux by

where

An essential ingredient of the HLL scheme are good estimates for the smallest and largest signal velocities. In the non-relativistic case, Einfeldt [48] proposed to calculate them based on the smallest and largest eigenvalues of Roe's matrix. This HLL scheme with Einfeldt's recipe is a very robust upwind scheme for the Euler equations and possesses the property of being positively conservative. The method is exact for single shocks, but it is very dissipative, especially at contact discontinuities.

Schneider et al. [161] have presented results in 1D ultra-relativistic hydrodynamics using a version of the HLL method with signal velocities given by

where is the relativistic sound speed, and where the bar denotes the arithmetic mean between the initial left and right states. Duncan & Hughes [46] have generalized this method to 2D SRHD and applied it to the simulation of relativistic extragalactic jets.

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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