

3.3 Two-shock approximation
for relativistic hydrodynamics
This approximate Riemann solver is obtained from a relativistic
extension of Colella’s method [57
] for classical fluid
dynamics, where it has been shown to handle shocks of arbitrary
strength [57, 300
]. In order to
construct Riemann solutions in the two-shock 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 Rankine-Hugoniot conditions across oblique
shocks. Balsara [13
] 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 and Woodward [64
] have developed a
similar Riemann solver based on the jump conditions across oblique
shocks making the solver more efficient.
Table 2 gives the converged solution for
the intermediate states obtained with both Balsara’s and Dai and
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 two-shock approximation should be tested in the
ultra-relativistic 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 two-shock
approximation for Riemann problems involving transversal velocities
still needs to be tested.

