

3.6 Relativistic HLL method
(RHLLE)
Schneider et al. [256
] 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. [256
] have presented
results in 1D relativistic hydrodynamics using a relativistic
version of the HLL method (RHLLE) 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 and Hughes [78
] have generalized
this method to 2D SRHD and applied it to the simulation of
relativistic extragalactic jets.

