Problem 1 was a demanding problem for relativistic hydrodynamic codes in the mid eighties [28, 75], while Problem 2 is a challenge even for today's state-of-the-art codes. The analytical solution of both problems can be obtained with program the RIEMANN (see Section 9.3).
Using artificial viscosity techniques, Centrella & Wilson  were able to reproduce the analytical solution with a 7% overshoot in , whereas Hawley et al.  got a 16% error in the shell density.
The results obtained with early relativistic SPH codes  were affected by systematic errors in the rarefaction wave and the constant states, large amplitude spikes at the contact discontinuity and large smearing. Smaller systematic errors and spikes are obtained with Laguna et al.'s (1993) code . This code also leads to a large overshoot in the shell's density. Much cleaner states are obtained with the methods of Chow & Monaghan (1997)  and Siegler & Riffert (1999) , both based on conservative formulations of the SPH equations. In the case of Chow & Monaghan's (1997) method , the spikes at the contact discontinuity disappear but at the cost of an excessive smearing. Shock profiles with relativistic SPH codes are more smeared out than with HRSC methods covering typically more than 10 zones.
Van Putten has considered a similar initial value problem with somewhat more extreme conditions (, ) and with a transversal magnetic field. For suitable choices of the smoothing parameters his results are accurate and stable, although discontinuities appear to be more smeared than with typical HRSC methods (6-7 zones for the strong shock wave; zones for the contact discontinuity).
An MPEG movie (Mov. 6) shows the Problem 1 blast wave evolution obtained with a modern HRSC method (the relativistic PPM method introduced in Section 3.1). The grid has 400 equidistant zones, and the relativistic shell is resolved by 16 zones. Because of both the high order accuracy of the method in smooth regions and its small numerical diffusion (the shock is resolved with 4-5 zones only) the density of the shell is accurately computed (errors less than 0.1%). Other codes based on relativistic Riemann solvers  give similar results (see Table 7). The relativistic HLL method  underestimates the density in the shell by about 10% in a 200 zone calculation.
|Centrella & Wilson
|1D||AV-mono||Stable profiles without oscillations. Velocity overestimated by 7%.|
|Hawley et al.
|1D||AV-mono||Stable profiles without oscillations. overestimated by 16%.|
|1D||FCT-lw||10-12 zones at the CD. Velocity overestimated by 4.5%.|
|1D||SPH-AV-0,1,2||Systematic errors in the rarefaction wave and the constant states. Large amplitude spikes at the CD. Excessive smearing at the shell.|
|Laguna et al.
|1D||SPH-AV-0||Large amplitude spikes at the CD. overestimated by 5%.|
|1D||van Putten||Stable profiles. Excessive smearing, specially at the CD ( zones).|
|Schneider et al.
|1D||SHASTA-c||Non monotonic intermediate states. underestimated by 10% with 200 zones.|
|Chow & Monaghan
|1D||SPH-RS-c||Stable profiles without spikes. Excessive smearing at the CD and at the shock.|
|Siegler & Riffert
|1D||SPH-cAV-c||Correct constant states. Large amplitude spikes at the CD. Excessive smearing at the shock transition ( zones).|
|1D||Roe-Eulderink||Correct with 500 zones. 4 zones in CD.|
|Schneider et al.
|1D||HLL-l||underestimated by 10% with 200 zones.|
|Martí & Müller
|1D||rPPM||Correct with 400 zones. 6 zones in CD.|
|Martí et al.
|1D, 2D||MFF-ppm||Correct with 400 zones. 6 zones in CD.|
|Wen et al.
|1D||rGlimm||No diffusion at discontinuities.|
|Yang et al.
|Donat et al.
|1D||MFF-eno||Correct with 400 zones. 8 zones in CD.|
|Aloy et al.
|3D||MFF-ppm||Correct with zones. 2 zones in CD.|
|Font et al.
|1D, 3D||MFF-l||Correct with 400 zones. 12-14 zones in CD.|
|1D, 3D||Roe type-l||Correct with 400 zones. 12-14 zones in CD.|
|1D, 3D||Flux split||overestimated by 5%. 8 zones in CD.|
Several HRSC methods based on relativistic Riemann solvers have used Problem 2 as a standard test [107, 106, 109, 55, 187, 43]. Table 8 gives a summary of the references where this test was considered.
|Norman & Winkler (1986) ||cAV-implicit||1.00|
|Dubal (1991) ||FCT-lw||0.80|
|Martí et al. (1991) ||Roe type-l||0.53|
|Marquina et al. (1992) ||LCA-phm||0.64|
|Martí & Müller (1996) ||rPPM||0.68|
|Falle & Komissarov (1996) ||Falle-Komissarov||0.47|
|Wen et al. (1997) ||rGlimm||1.00|
|Chow & Monaghan (1997) ||SPH-RS-c||1.16|
|Donat et al. (1998) ||MFF-phm||0.60|
An MPEG movie (Mov. 7) shows the Problem 2 blast wave evolution obtained with the relativistic PPM method introduced in Section 3.1 on a grid of 2000 equidistant zones. At this resolution the relativistic PPM code yields a converged solution. The method of Falle & Komissarov  requires a seven-level adaptive grid calculation to achieve the same, the finest grid spacing corresponding to a grid of 3200 zones. As their code is free of numerical diffusion and dispersion, Wen et al.  are able to handle this problem with high accuracy (see Fig 8). At lower resolution (400 zones) the relativistic PPM method only reaches 69% of the theoretical shock compression value (54% in case of the second-order accurate upwind method of Falle & Komissarov ; 60% with the code of Donat et al. ).
Chow & Monaghan  have considered Problem 2 to test their relativistic SPH code. Besides a 15% overshoot in the shell's density, the code produces a non-causal blast wave propagation speed (i.e., ).
The initial data corresponding to this test, consisting of three constant states with large pressure jumps at the discontinuities separating the states (at x = 0.1 and x = 0.9), as well as the properties of the blast waves created by the decay of the initial discontinuities, are listed in Table 9 . The propagation velocity of the two blast waves is slower than in the Newtonian case, but very close to the speed of light (0.9776 and -0.9274 for the shock wave propagating to the right and left, respectively). Hence, the shock interaction occurs later (at t = 0.420) than in the Newtonian problem (at t = 0.028). The top panel in Fig. 9 shows four snapshots of the density distribution, including the moment of the collision of the blast waves at t = 0.420 and x = 0.5106. At the time of collision the two shells have a width of (left shell) and (right shell), respectively, i.e., the whole interaction takes place in a very thin region (about 10 times smaller than in the Newtonian case, where ).
The collision gives rise to a narrow region of very high density (see lower panel of Fig. 9), bounded by two shocks moving at speeds 0.088 (shock at the left) and 0.703 (shock at the right) and large compression ratios (7.26 and 12.06, respectively) well above the classical limit for strong shocks (6.0 for ). The solution just described applies until t = 0.430 when the next interaction takes place.
The complete analytical solution before and after the collision up to time t = 0.430 can be obtained following Appendix II in .
An MPEG movie (Mov. 10) shows the evolution of the density up to the time of shock collision at t = 0.4200. The movie was obtained with the relativistic PPM code of Martí & Müller . The presence of very narrow structures with large density jumps requires very fine zoning to resolve the states properly. For the movie a grid of 4000 equidistant zones was used. The relative error in the density of the left (right) shell is always less than 2.0% (0.6%), and is about 1.0% (0.5%) at the moment of shock collision. Profiles obtained with the relativistic Godunov method (first-order accurate, not shown) show relative errors in the density of the left (right) shell of about 50% (16%) at t = 0.20. The errors drop only slightly to about 40% (5%) at the time of collision (t = 0.420).
An MPEG movie (Mov. 11) shows the numerical solution after the interaction has occurred. Compared to the other MPEG movie (Mov. 10) a very different scaling for the x -axis had to be used to display the narrow dense new states produced by the interaction. Obviously, the relativistic PPM code resolves the structure of the collision region satisfactorily well, the maximum relative error in the density distribution being less than 2.0%. When using the first-order accurate Godunov method instead, the new states are strongly smeared out and the positions of the leading shocks are wrong.
|Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
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