### 3.6 Relativistic HLL method (RHLLE)

Schneider et al. [257] have proposed to use the method of Harten, Lax and van Leer (HLL
hereafter [122]) to integrate the equations of SRHD. This method avoids the explicit calculation of the
eigenvalues and eigenvectors of the Jacobian matrices and is based on an approximate solution of the
original Riemann problems with a single intermediate state
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 [81] 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. [257] have presented results in 1D relativistic hydrodynamics using a relativistic version
of the HLL method (RHLLE) with signal velocities given by

where c_{s} is the relativistic sound speed, and where the bar denotes the arithmetic mean between the initial
left and right states. Duncan and Hughes [78] have generalized this method to 2D SRHD and applied it to
the simulation of relativistic extragalactic jets.