3.4 Roe-type relativistic solvers3 High-Resolution Shock-Capturing Methods3.2 The relativistic Glimm method

3.3 Two-shock approximation for relativistic hydrodynamics 

This approximate Riemann solver is obtained from a relativistic extension of Colella's method [31Jump To The Next Citation Point In The Article] for classical fluid dynamics, where it has been shown to handle shocks of arbitrary strength [31, 191Jump To The Next Citation Point In The Article]. 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 [8Jump To The Next Citation Point In The Article] 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 [36Jump To The Next Citation Point In The Article] have developed a similar Riemann solver based on the jump conditions across oblique shocks making the solver more efficient.

  

table387

Table 1: Pressure tex2html_wrap_inline5597, velocity tex2html_wrap_inline5599, and densities tex2html_wrap_inline5601 (left), tex2html_wrap_inline5603 (right) for the intermediate state obtained for the two-shock approximation of Balsara [8Jump To The Next Citation Point In The Article] (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 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.



3.4 Roe-type relativistic solvers3 High-Resolution Shock-Capturing Methods3.2 The relativistic Glimm method

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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