6.2 Propagation of relativistic blast 6 Test Bench6 Test Bench

6.1 Relativistic shock heating in planar, cylindrical, and spherical geometry 

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 ([18] in planar symmetry; [111Jump To The Next Citation Point In The Article] 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., tex2html_wrap_inline6197) gas with coordinate velocity tex2html_wrap_inline6199 and Lorentz factor tex2html_wrap_inline6201 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 (tex2html_wrap_inline6203). Due to conservation of energy across the shock the gas has a specific internal energy given by

equation1212

The compression ratio of shocked and unshocked gas, tex2html_wrap_inline6205, follows from

equation1214

where tex2html_wrap_inline5745 is the adiabatic index of the equation of state. The shock velocity is given by

equation1220

In the unshocked region (tex2html_wrap_inline6209) the pressure-less gas flow is self-similar and has a density distribution given by

equation1224

where tex2html_wrap_inline6211 for planar, cylindrical or spherical geometry, and where tex2html_wrap_inline6213 is the density of the inflowing gas at infinity (see Fig.  3).

  

Click on thumbnail to view image

Figure 3: Schematic solution of the shock heating problem in spherical geometry. The initial state consists of a spherically symmetric flow of cold ( p=0) gas of unit rest mass density having a coordinate inflow velocity tex2html_wrap_inline5607 everywhere. A shock is generated at the center of the sphere, which propagates upstream with constant speed. The post-shock state is constant and at rest. The pre-shock state, where the flow is self-similar, has a density which varies as tex2html_wrap_inline5609 with time t and radius r .

In the Newtonian case the compression ratio tex2html_wrap_inline6205 of shocked and unshocked gas cannot exceed a value of tex2html_wrap_inline6227 independently of the inflow velocity. This is different for relativistic flows, where tex2html_wrap_inline6205 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 tex2html_wrap_inline6205 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.

 

References                 tex2html_wrap_inline6233   Method tex2html_wrap_inline5621   tex2html_wrap_inline5623 [%]
Centrella & Wilson
(1984) [28Jump To The Next Citation Point In The Article]
0 AV-mono 2.29 tex2html_wrap_inline6239
Hawley et al.
(1984) [75Jump To The Next Citation Point In The Article]
0 AV-mono 4.12 tex2html_wrap_inline6239
Norman & Winkler
(1986) [131Jump To The Next Citation Point In The Article
0 cAV-implicit 10.0 0.01
McAbee et al.
(1989) [113Jump To The Next Citation Point In The Article]
0 AV-mono 10.0 2.6
Martí et al.
(1991) [107Jump To The Next Citation Point In The Article]
0 Roe type-l 23 0.2
Marquina et al.
(1992) [106Jump To The Next Citation Point In The Article]
0 LCA-phm 70 0.1
Eulderink
(1993) [49Jump To The Next Citation Point In The Article]
0 Roe-Eulderink 625 tex2html_wrap_inline6243 Popup Footnote
Schneider et al.
(1993) [161Jump To The Next Citation Point In The Article]
0 HLL-l tex2html_wrap_inline6245 0.2 Popup Footnote
0 SHASTA-c tex2html_wrap_inline6245 0.5 Popup Footnote
Dolezal & Wong
(1995) [42Jump To The Next Citation Point In The Article]
0 LCA-eno tex2html_wrap_inline6253   tex2html_wrap_inline6255 0.1 Popup Footnote
Martí & Müller
(1996) [109Jump To The Next Citation Point In The Article]
0 rPPM 224 0.03
Falle & Komissarov
(1996) [55Jump To The Next Citation Point In The Article]
0 Falle-Komissarov  224 tex2html_wrap_inline6255 0.1 Popup Footnote
Romero et al.
(1996) [157Jump To The Next Citation Point In The Article]
2 Roe type-l 2236 2.2
Martí et al.
(1997) [111Jump To The Next Citation Point In The Article]
1 MFF-ppm 70 1.0
Chow & Monaghan
(1997) [30Jump To The Next Citation Point In The Article]
0 SPH-RS-c 70 0.2
Wen et al.
(1997) [187Jump To The Next Citation Point In The Article]
2 rGlimm 224 tex2html_wrap_inline6263
Donat et al.
(1998) [43Jump To The Next Citation Point In The Article]
0 MFF-eno 224 tex2html_wrap_inline6255 0.1 Popup Footnote
Aloy et al.
(1999) [3Jump To The Next Citation Point In The Article]
0 MFF-ppm tex2html_wrap_inline6269 3.5 Popup Footnote
Sieglert & Riffert
(1999) [164Jump To The Next Citation Point In The Article]
0 SPH-cAV-c 1000 tex2html_wrap_inline6255 0.1 Popup Footnote
  
Table 4: Summary of relativistic shock heating test calculations by various authors in planar (tex2html_wrap_inline5615), cylindrical (tex2html_wrap_inline5617), and spherical (tex2html_wrap_inline5619) geometry. tex2html_wrap_inline5621 and tex2html_wrap_inline5623 are the maximum inflow Lorentz factor and compression ratio error extracted from tables and figures of the corresponding reference. tex2html_wrap_inline5621 should only be considered as indicative of the maximum Lorentz factor achievable by every method. Methods are described in Sections  3 and 4 and their basic properties summarized in Section  5 (Tables  2, 3).

Explicit finite-difference techniques based on a non-conservative formulation of the hydrodynamic equations and on non-consistent artificial viscosity [28Jump To The Next Citation Point In The Article, 75Jump To The Next Citation Point In The Article] are able to handle flow Lorentz factors up to tex2html_wrap_inline6239 with moderately large errors (tex2html_wrap_inline6289) at best [190, 113]. Norman & Winkler [131Jump To The Next Citation Point In The Article] got very good results (tex2html_wrap_inline6291) 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 [107Jump To The Next Citation Point In The Article, 106Jump To The Next Citation Point In The Article, 49Jump To The Next Citation Point In The Article, 161Jump To The Next Citation Point In The Article, 50Jump To The Next Citation Point In The Article, 109Jump To The Next Citation Point In The Article, 55Jump To The Next Citation Point In The Article]. 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 [42Jump To The Next Citation Point In The Article, 187Jump To The Next Citation Point In The Article, 3Jump To The Next Citation Point In The Article].

Schneider et al. [161Jump To The Next Citation Point In The Article] 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 tex2html_wrap_inline6205 was reduced by a factor of two when using HLL.

Within SPH methods, Chow & Monaghan [30Jump To The Next Citation Point In The Article] have obtained results comparable to those of HRSC methods (tex2html_wrap_inline6295) for flow Lorentz factors up to 70, using a relativistic SPH code with Riemann solver guided dissipation. Sieglert & Riffert [164Jump To The Next Citation Point In The Article] 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. [164Jump To The Next Citation Point In The Article]; 20-24 in the code of ref. [30Jump To The Next Citation Point In The Article]) 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 tex2html_wrap_inline5627 (tex2html_wrap_inline6303). These results are obtained with the relativistic PPM code of [109Jump To The Next Citation Point In The Article], which uses an exact Riemann solver based on the procedure described in Section  2.3 .

  

Click on thumbnail to view movie

Figure 4: MPEG movie showing the evolution of the density distribution for the shock heating problem with an inflow velocity tex2html_wrap_inline5627 in Cartesian coordinates. The reflecting wall is located at x=0. The adiabatic index of the gas is 4/3. For numerical reasons, the specific internal energy of the inflowing cold gas is set to a small finite value (tex2html_wrap_inline5631). The final frame of the movie also shows the analytical solution (blue lines). The simulation has been performed on an equidistant grid of 100 zones.

The shock wave is resolved by three zones and there are no post-shock numerical oscillations. The density increases by a factor tex2html_wrap_inline6311 across the shock. Near x =0 the density distribution slightly undershoots the analytical solution (by tex2html_wrap_inline6315) 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 tex2html_wrap_inline6317, 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 [157Jump To The Next Citation Point In The Article, 111Jump To The Next Citation Point In The Article, 187Jump To The Next Citation Point In The Article]. Aloy et al. [3Jump To The Next Citation Point In The Article] 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 [130Jump To The Next Citation Point In The Article], is a numerical artifact which is considerably reduced when Marquina's scheme is used [44]. In passing we note that the strong overheating found by Noh [130Jump To The Next Citation Point In The Article] for the spherical shock reflection test using PPM (Fig. 24 in [130]) 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 [157Jump To The Next Citation Point In The Article]).



6.2 Propagation of relativistic blast 6 Test Bench6 Test Bench

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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