Two-shock approximation for relativistic hydrodynamics

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.

Table 2: Pressure $p∗$ , velocity $v∗$ , and densities $ρL ∗$ (left), $ρR∗$ (right) for the intermediate state obtained for the two-shock approximation of Balsara (B) [13

] and of Dai and Woodward (DW) [64 ] compared to the exact solution (Exact) for the Riemann problems defined in Section 6.2.

	Method	$p∗$	$v∗$	$ρL∗$	$ρR∗$
	B	1.440 × 10⁺⁰	7.131 × 10^–1	2.990 × 10⁺⁰	5.069 × 10⁺⁰
Problem 1	DW	1.440 × 10⁺⁰	7.131 × 10^–1	2.990 × 10⁺⁰	5.066 × 10⁺⁰
	Exact	1.445 × 10⁺⁰	7.137 × 10^–1	2.640 × 10⁺⁰	5.062 × 10⁺⁰
	B	1.543 × 10⁺¹	9.600 × 10^–1	7.325 × 10^–2	1.709 × 10⁺¹
Problem 2	DW	1.513 × 10⁺¹	9.608 × 10^–1	7.254 × 10^–2	1.742 × 10⁺¹
	Exact	1.293 × 10⁺¹	9.546 × 10^–1	3.835 × 10^–2	1.644 × 10⁺¹