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.
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Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |