This approximate Riemann solver is obtained from a relativistic
extension of Colella's method [31] for classical fluid dynamics, where it has been shown to handle
shocks of arbitrary strength [31,
191]. In order to construct Riemann solutions in the twoshock
approximation one analytically continues shock waves towards the
rarefaction side (if present) of the zone interface instead of
using an actual rarefaction wave solution. Thereby one gets rid
of the coupling of the normal and tangential components of the
flow velocity (see Section
2.3), and the remaining minor algebraic complications are the
RankineHugoniot conditions across oblique shocks. Balsara [8] has developed an approximate relativistic Riemann solver of
this kind by solving the jump conditions in the shocks' rest
frames in the absence of transverse velocities, after appropriate
Lorentz transformations. Dai & Woodward [36] have developed a similar Riemann solver based on the jump
conditions across oblique shocks making the solver more
efficient.
Table 1:
Pressure
, velocity
, and densities
(left),
(right) for the intermediate state obtained for the twoshock
approximation of Balsara [8] (B) and of Dai & Woodward [36] (DW) compared to the exact solution (Exact) for the Riemann
problems defined in Section
6.2
.
Table
1
gives the converged solution for the intermediate states
obtained with both Balsara's and Dai & Woodward's procedure
for the case of the Riemann problems defined in Section
6.2
(involving strong rarefaction waves) together with the exact
solution. Despite the fact that both approximate methods involve
very different algebraic expressions, their results differ by
less than 2%. However, the discrepancies are much larger when
compared with the exact solution (up to a 100% error in the
density of the left intermediate state in Problem 2). The
accuracy of the twoshock approximation should be tested in the
ultrarelativistic limit, where the approximation can produce
large errors in the Lorentz factor (in the case of Riemann
problems involving strong rarefaction waves) with important
implications for the fluid dynamics. Finally, the suitability of
the twoshock approximation for Riemann problems involving
transversal velocities still needs to be tested.

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr19993
© MaxPlanckGesellschaft. ISSN 14338351
Problems/Comments to
livrev@aeipotsdam.mpg.de
