Shock heating of a cold fluid in planar, cylindrical or spherical geometry has been used since the early developments of numerical relativistic hydrodynamics as a test case for hydrodynamic codes, because it has an analytical solution ( in planar symmetry;  in cylindrical and spherical symmetry), and because it involves the propagation of a strong relativistic shock wave.
In planar geometry, an initially homogeneous, cold (i.e., ) gas with coordinate velocity and Lorentz factor is supposed to hit a wall, while in the case of cylindrical and spherical geometry the gas flow converges towards the axis or the center of symmetry. In all three cases the reflection causes compression and heating of the gas as kinetic energy is converted into internal energy. This occurs in a shock wave, which propagates upstream. Behind the shock the gas is at rest (). Due to conservation of energy across the shock the gas has a specific internal energy given by
The compression ratio of shocked and unshocked gas, , follows from
where is the adiabatic index of the equation of state. The shock velocity is given by
In the unshocked region () the pressure-less gas flow is self-similar and has a density distribution given by
where for planar, cylindrical or spherical geometry, and where is the density of the inflowing gas at infinity (see Fig. 3).
In the Newtonian case the compression ratio of shocked and unshocked gas cannot exceed a value of independently of the inflow velocity. This is different for relativistic flows, where grows linearly with the flow Lorentz factor and becomes infinite as the inflowing gas velocity approaches to speed of light.
The maximum flow Lorentz factor achievable for a hydrodynamic code with acceptable errors in the compression ratio is a measure of the code's quality. Table 4 contains a summary of the results obtained for the shock heating test by various authors.
|Centrella & Wilson
|Hawley et al.
|Norman & Winkler
|McAbee et al.
|Martí et al.
|Marquina et al.
|Schneider et al.
|Dolezal & Wong
|Martí & Müller
|Falle & Komissarov
|Romero et al.
|Martí et al.
|Chow & Monaghan
|Wen et al.
|Donat et al.
|Aloy et al.
|Sieglert & Riffert
Explicit finite-difference techniques based on a non-conservative formulation of the hydrodynamic equations and on non-consistent artificial viscosity [28, 75] are able to handle flow Lorentz factors up to with moderately large errors () at best [190, 113]. Norman & Winkler  got very good results () for a flow Lorentz factor of 10 using consistent artificial viscosity terms and an implicit adaptive-mesh method.
The performance of explicit codes improved significantly when numerical methods based on Riemann solvers were introduced [107, 106, 49, 161, 50, 109, 55]. For some of these codes the maximum flow Lorentz factor is only limited by the precision by which numbers are represented on the computer used for the simulation [42, 187, 3].
Schneider et al.  have compared the accuracy of a code based on the relativistic HLL Riemann solver with different versions of relativistic FCT codes for inflow Lorentz factors in the range 1.6 to 50. They found that the error in was reduced by a factor of two when using HLL.
Within SPH methods, Chow & Monaghan  have obtained results comparable to those of HRSC methods () for flow Lorentz factors up to 70, using a relativistic SPH code with Riemann solver guided dissipation. Sieglert & Riffert  have succeeded in reproducing the post-shock state accurately for inflow Lorentz factors of 1000 with a code based on a consistent formulation of artificial viscosity. However, the dissipation introduced by SPH methods at the shock transition is very large (10-12 particles in the code of ref. ; 20-24 in the code of ref. ) compared with the typical dissipation of HRSC methods (see below).
The performance of a HRSC method based on a relativistic Riemann solver is illustrated by means of an MPEG movie (Mov. 4) for the planar shock heating problem for an inflow velocity (). These results are obtained with the relativistic PPM code of , which uses an exact Riemann solver based on the procedure described in Section 2.3 .
The shock wave is resolved by three zones and there are no post-shock numerical oscillations. The density increases by a factor across the shock. Near x =0 the density distribution slightly undershoots the analytical solution (by ) due to the numerical effect of wall heating. The profiles obtained for other inflow velocities are qualitatively similar. The mean relative error of the compression ratio , and, in agreement with other codes based on a Riemann solver, the accuracy of the results does not exhibit any significant dependence on the Lorentz factor of the inflowing gas.
Some authors have considered the problem of shock heating in cylindrical or spherical geometry using adapted coordinates to test the numerical treatment of geometrical factors [157, 111, 187]. Aloy et al.  have considered the spherically symmetric shock heating problem in 3D Cartesian coordinates as a test case for both the directional splitting and the symmetry properties of their code GENESIS. The code is able to handle this test up to inflow Lorentz factors of the order of 700.
In the shock reflection test conventional schemes often give numerical approximations which exhibit a consistent O (1) error for the density and internal energy in a few cells near the reflecting wall. This 'overheating', as it is known in classical hydrodynamics , is a numerical artifact which is considerably reduced when Marquina's scheme is used . In passing we note that the strong overheating found by Noh  for the spherical shock reflection test using PPM (Fig. 24 in ) is not a problem of PPM, but of his implementation of PPM. When properly implemented PPM gives a density undershoot near the origin of about 9% in case of a non-relativistic flow. PLM gives an undershoot of 14% in case of ultra-relativistic flows (e.g., Tab. 1 and Fig. 1 in ).
|Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
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