7 Applications6 Test Bench6.1 Relativistic shock heating in

6.2 Propagation of relativistic blast waves 

Riemann problems with large initial pressure jumps produce blast waves with dense shells of material propagating at relativistic speeds (see Fig.  5). For appropriate initial conditions, both the speed of the leading shock front and the velocity of the shell material approach the speed of light producing very narrow structures. The accurate description of these thin, relativistic shells involving large density contrasts is a challenge for any numerical code. Some particular blast wave problems have become standard numerical tests. Here we consider the two most common of these tests. The initial conditions are given in Table  5 .

  

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Figure 5: Generation and propagation of a relativistic blast wave (schematic). The large pressure jump at a discontinuity initially located at r=0.5 gives rise to a blast wave and a dense shell of material propagating at relativistic speeds. For appropriate initial conditions both the speed of the leading shock front and the velocity of the shell approach the speed of light producing very narrow structures.

Problem 1 was a demanding problem for relativistic hydrodynamic codes in the mid eighties [28Jump To The Next Citation Point In The Article, 75Jump To The Next Citation Point In The Article], 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).

   table1630

Table 5: Initial data (pressure p, density tex2html_wrap_inline5637, velocity v) for two common relativistic blast wave test problems. The decay of the initial discontinuity leads to a shock wave (velocity tex2html_wrap_inline5641, compression ratio tex2html_wrap_inline5643) and the formation of a dense shell (velocity tex2html_wrap_inline5645, time-dependent width tex2html_wrap_inline5647) both propagating to the right. The gas is assumed to be ideal with an adiabatic index tex2html_wrap_inline5649 .

6.2.1 Problem 1 

In Problem 1, the decay of the initial discontinuity gives rise to a dense shell of matter with velocity tex2html_wrap_inline6357 (tex2html_wrap_inline6359) propagating to the right. The shell trailing a shock wave of speed tex2html_wrap_inline6361 increases its width, tex2html_wrap_inline5647, according to tex2html_wrap_inline6365, i.e., at time t = 0.4 the shell covers about 4% of the grid (tex2html_wrap_inline6369). Tables  6 and  7 give a summary of the references where this test was considered for non-HRSC and HRSC methods, respectively.

Using artificial viscosity techniques, Centrella & Wilson [28Jump To The Next Citation Point In The Article] were able to reproduce the analytical solution with a 7% overshoot in tex2html_wrap_inline5645, whereas Hawley et al. [75Jump To The Next Citation Point In The Article] got a 16% error in the shell density.

The results obtained with early relativistic SPH codes [102Jump To The Next Citation Point In The Article] 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 [89Jump To The Next Citation Point In The Article]. 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) [30Jump To The Next Citation Point In The Article] and Siegler & Riffert (1999) [164Jump To The Next Citation Point In The Article], both based on conservative formulations of the SPH equations. In the case of Chow & Monaghan's (1997) method [30Jump To The Next Citation Point In The Article], 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 (tex2html_wrap_inline6373, tex2html_wrap_inline6375) 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; tex2html_wrap_inline6379 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 [50Jump To The Next Citation Point In The Article] give similar results (see Table  7). The relativistic HLL method [161Jump To The Next Citation Point In The Article] underestimates the density in the shell by about 10% in a 200 zone calculation.

  

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Figure 6: MPEG movie showing the evolution of the density distribution for the relativistic blast wave Problem 1 (defined in Table  5). The final frame of the movie also shows the analytical solution (blue lines). The simulation has been performed with relativistic PPM on an equidistant grid of 400 zones.
References                 Dim. Method Comments
Centrella & Wilson
(1984) [28]
1D AV-mono Stable profiles without oscillations. Velocity overestimated by 7%.
Hawley et al.
(1984) [75Jump To The Next Citation Point In The Article]
1D AV-mono Stable profiles without oscillations. tex2html_wrap_inline6381 overestimated by 16%.
Dubal
(1991) [45Jump To The Next Citation Point In The Article]
1D FCT-lw 10-12 zones at the CD. Velocity overestimated by 4.5%.
Mann
(1991) [102Jump To The Next Citation Point In The Article]
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.
(1993) [89]
1D SPH-AV-0 Large amplitude spikes at the CD. tex2html_wrap_inline6381 overestimated by 5%.
van Putten
(1993) [181Jump To The Next Citation Point In The Article]
1D van Putten Stable profiles. Excessive smearing, specially at the CD (tex2html_wrap_inline6379 zones).
Schneider et al.
(1993) [161Jump To The Next Citation Point In The Article]
1D SHASTA-c Non monotonic intermediate states. tex2html_wrap_inline6381 underestimated by 10% with 200 zones.
Chow & Monaghan
(1997) [30Jump To The Next Citation Point In The Article]
1D SPH-RS-c Stable profiles without spikes. Excessive smearing at the CD and at the shock.
Siegler & Riffert
(1999) [164Jump To The Next Citation Point In The Article]
1D SPH-cAV-c Correct constant states. Large amplitude spikes at the CD. Excessive smearing at the shock transition (tex2html_wrap_inline6389 zones).
  
Table 6: Non-HRSC methods - Summary of references where the blast wave Problem 1 (defined in Table  5) has been considered in 1D, 2D, and 3D, respectively. The methods are described in Sections  3 and 4 and their basic properties summarized in Section  5 (Tables  2, 3). Note: CD stands for contact discontinuity.
References                 Dim.    Method Comments Popup Footnote
Eulderink
(1993) [49Jump To The Next Citation Point In The Article]
1D Roe-Eulderink Correct tex2html_wrap_inline6381 with 500 zones. 4 zones in CD.
Schneider et al.
(1993) [161Jump To The Next Citation Point In The Article]
1D HLL-l tex2html_wrap_inline6381 underestimated by 10% with 200 zones.
Martí & Müller
(1996) [109Jump To The Next Citation Point In The Article]
1D rPPM Correct tex2html_wrap_inline6381 with 400 zones. 6 zones in CD.
Martí et al.
(1997) [111Jump To The Next Citation Point In The Article]
1D, 2D MFF-ppm Correct tex2html_wrap_inline6381 with 400 zones. 6 zones in CD.
Wen et al.
(1997) [187Jump To The Next Citation Point In The Article]
1D rGlimm No diffusion at discontinuities.
Yang et al.
(1997) [194Jump To The Next Citation Point In The Article]
1D rBS Stable profiles.
Donat et al.
(1998) [43Jump To The Next Citation Point In The Article]
1D MFF-eno Correct tex2html_wrap_inline6381 with 400 zones. 8 zones in CD.
Aloy et al.
(1999) [3Jump To The Next Citation Point In The Article]
3D MFF-ppm Correct tex2html_wrap_inline6381 with tex2html_wrap_inline6403 zones. 2 zones in CD.
Font et al.
(1999) [59Jump To The Next Citation Point In The Article]
1D, 3D MFF-l Correct tex2html_wrap_inline6381 with 400 zones. 12-14 zones in CD.
1D, 3D Roe type-l Correct tex2html_wrap_inline6381 with 400 zones. 12-14 zones in CD.
1D, 3D Flux split tex2html_wrap_inline6381 overestimated by 5%. 8 zones in CD.
  
Table 7: HRSC methods - Summary of references where the blast wave Problem 1 (defined in Table  5) has been considered in 1D, 2D, and 3D, respectively. The methods are described in Sections  3 and 4 and their basic properties summarized in Section  5 (Tables  2, 3). Note: CD stands for contact discontinuity.

6.2.2 Problem 2 

Problem 2 was first considered by Norman & Winkler [131Jump To The Next Citation Point In The Article]. The flow pattern is similar to that of Problem 1, but more extreme. Relativistic effects reduce the post-shock state to a thin dense shell with a width of only about 1% of the grid length at t = 0.4. The fluid in the shell moves with tex2html_wrap_inline6413 (i.e., tex2html_wrap_inline6415), while the leading shock front propagates with a velocity tex2html_wrap_inline6417 (i.e., tex2html_wrap_inline6419). The jump in density in the shell reaches a value of 10.6. Norman & Winkler [131Jump To The Next Citation Point In The Article] obtained very good results with an adaptive grid of 400 zones using an implicit hydro-code with artificial viscosity. Their adaptive grid algorithm placed 140 zones of the available 400 zones within the blast wave thereby accurately capturing all features of the solution.

Several HRSC methods based on relativistic Riemann solvers have used Problem 2 as a standard test [107Jump To The Next Citation Point In The Article, 106Jump 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, 187Jump To The Next Citation Point In The Article, 43Jump To The Next Citation Point In The Article]. Table  8 gives a summary of the references where this test was considered.

References Method tex2html_wrap_inline6421
Norman & Winkler (1986) [131Jump To The Next Citation Point In The Article cAV-implicit 1.00
Dubal (1991) Popup Footnote  [45Jump To The Next Citation Point In The Article] FCT-lw 0.80
Martí et al. (1991) [107Jump To The Next Citation Point In The Article] Roe type-l 0.53
Marquina et al. (1992) [106] LCA-phm 0.64
Martí & Müller (1996) [109Jump To The Next Citation Point In The Article] rPPM 0.68
Falle & Komissarov (1996) [55Jump To The Next Citation Point In The Article] Falle-Komissarov  0.47
Wen et al. (1997) [187Jump To The Next Citation Point In The Article] rGlimm 1.00
Chow & Monaghan (1997) [30Jump To The Next Citation Point In The Article] SPH-RS-c 1.16 Popup Footnote
Donat et al. (1998) [43Jump To The Next Citation Point In The Article] MFF-phm 0.60
  
Table 8: Summary of references where the blast wave Problem 2 (defined in Table  5) has been considered. The methods are described in Sections  3 and 4 and their basic properties summarized in Section  5 (Tables  2, 3).

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 [55Jump To The Next Citation Point In The Article] 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. [187Jump To The Next Citation Point In The Article] 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 [55]; 60% with the code of Donat et al. [43Jump To The Next Citation Point In The Article]).

  

Click on thumbnail to view movie

Figure 7: MPEG movie showing the evolution of the density distribution for the relativistic blast wave Problem 2 (defined in Table  5). The final frame of the movie also shows the analytical solution (blue lines). The simulation has been performed with relativistic PPM on an equidistant grid of 2000 zones.

  

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Figure 8: Results from [187Jump To The Next Citation Point In The Article] for the relativistic blast wave Problems 1 (left column) and 2 (right column), respectively. The relativistic Glimm method is only used in regions with steep gradients. Standard finite difference schemes are applied in the smooth remaining part of the computational domain. In the above plots, Lax and LW stand respectively for Lax and Lax-Wendroff methods; G refers to the pure Glimm method.

Chow & Monaghan [30Jump To The Next Citation Point In The Article] 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., tex2html_wrap_inline6425).

6.2.3 Collision of two relativistic blast waves 

The collision of two strong blast waves was used by Woodward & Colella [191] to compare the performance of several numerical methods in classical hydrodynamics. In the relativistic case, Yang et al. [194] considered this problem to test the high-order extensions of the relativistic beam scheme, whereas Martí & Müller [109Jump To The Next Citation Point In The Article] used it to evaluate the performance of their relativistic PPM code. In this last case, the original boundary conditions were changed (from reflecting to outflow) to avoid the reflection and subsequent interaction of rarefaction waves, allowing for a comparison with an analytical solution. In the following we summarize the results on this test obtained by Martí & Müller in [109Jump To The Next Citation Point In The Article].
   table2277

Table 9: Initial data (pressure p, density tex2html_wrap_inline5637, velocity v) for the test problem of two colliding relativistic blast waves. The decay of the initial discontinuities (at x = 0.1 and x = 0.9) produces two shock waves (velocities tex2html_wrap_inline5641, compression ratios tex2html_wrap_inline5643) moving in opposite directions followed by two trailing dense shells (velocities tex2html_wrap_inline5645, time-dependent widths tex2html_wrap_inline5647). The gas is assumed to be ideal with an adiabatic index tex2html_wrap_inline5669 .

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 tex2html_wrap_inline6479 (left shell) and tex2html_wrap_inline6481 (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 tex2html_wrap_inline6483).

  

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Figure 9: The top panel shows a sequence of snapshots of the density profile for the colliding relativistic blast wave problem up to the moment when the waves begin to interact. The density profile of the new states produced by the interaction of the two waves is shown in the bottom panel (note the change in scale on both axes with respect to the top panel).

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 tex2html_wrap_inline5669). 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 [109Jump To The Next Citation Point In The Article].

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 [109Jump To The Next Citation Point In The Article]. 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.

  

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Figure 10: MPEG movie showing the evolution of the density distribution for the colliding relativistic blast wave problem up to the interaction of the waves. The final frame of the movie also shows the analytical solution (blue lines). The computation has been performed with relativistic PPM on an equidistant grid of 4000 zones.

  

Click on thumbnail to view movie

Figure 11: MPEG movie showing the evolution of the density distribution for the colliding relativistic blast wave problem around the time of interaction of the waves at an enlarged spatial scale. The final frame of the movie also shows the analytical solution (blue lines). The computation has been performed with relativistic PPM on an equidistant grid of 4000 zones.


7 Applications6 Test Bench6.1 Relativistic shock heating in

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
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