Propagation of relativistic blast waves

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 Figure 7

). 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 7.

Figure 7: 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 [50 , 123 ], 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 RIEMANN (see Section 9.4).

Table 7: Initial data (pressure $p$ , density $ρ$ , velocity $v$ ) for two common relativistic blast wave test problems. The decay of the initial discontinuity leads to a shock wave (velocity $v shock$ , compression ratio $σshock$ ) and the formation of a dense shell (velocity $vshell$ , time-dependent width $wshell$ ) both propagating to the right. The gas is assumed to be ideal with an adiabatic index $γ$ = 5/3.

		Problem 1			Problem 2
	Left		Right	Left		Right
$p$	13.33		0.00	1000.00		0.01
$ρ$	10.00		1.00	1.00		1.00
$v$	0.00		0.00	0.00		0.00
$vshell$		0.72			0.960
$wshell$		0.11 t			0.026 t
$vshock$		0.83			0.986
$σshock$		5.07			10.75

6.2.1 Problem 1

In Problem 1, the decay of the initial discontinuity gives rise to a dense shell of matter with velocity $vshell$ = 0.72 ( $Wshell$ = 1.38) propagating to the right. The shell trailing a shock wave of speed $vshock$ = 0.83 increases its width $wshell$ according to $wshell$ = 0.11 t, i.e., at time t = 0.4 the shell covers about 4% of the grid ( $0 ≤ x ≤ 1$ ). Tables 8 and 9 give a summary of the references where this test was considered for non-HRSC and HRSC methods, respectively.

Table 8: Summary of references where the blast wave problem 1 (defined in Table 7) has been considered in 1D, 2D and, 3D, respectively. Methods are described in Sections 3 and 4, and their basic properties are summarized in Section 5 (Tables 3, 4, and 5). Note that CD stands for contact discontinuity.

References	Dim.	Method	Comments
Centrella and Wilson (1984) [50 ]	1D	AV-mono	Stable profiles without oscillations;
			velocity overestimated by 7%.

Hawley et al. (1984) [123 ]	1D	AV-mono	Stable profiles without oscillations;
			$ρshell$ overestimated by 16%.

Dubal (1991)¹⁰ [77 ]	1D	FCT-lw	10–12 zones at the CD;
			velocity overestimated by 4.5%.

Mann (1991) [172 ]	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) [150 ]	1D	SPH-AV-0	Large amplitude spikes at the CD;
			$ρshell$ overestimated by 5%.

van Putten (1993)¹¹ [287 ]	1D	van Putten	Stable profiles;
			excessive smearing, especially of the
			CD ( $≈$ 50 zones).

Schneider et al. (1993) [257 ]	1D	SHASTA-c	Non-monotonic intermediate states;
			$ρshell$ underestimated by 10% with
			200 zones.

Chow and Monaghan (1997) [53 ]	1D	SPH-RS-c	Monotonic profiles;
			excessive smearing of CD and shock.

Siegler and Riffert (1999) [262 ]	1D	SPH-cAV-c	Correct constant states;
			large amplitude spikes at the CD;
			excessive smearing of shock.

Muir (2002) [204 ]	1D, 3D	SPH-RS-gr	Monotonic profiles;
			excessive smearing of CD and shock.

Anninos and Fragile (2002) [10 ]	1D, 3D	cAV-mono	Stable profiles without oscillations;
			correct constant states.

Table 9: Summary of references where the blast wave Problem 1 (defined in Table 7) has been considered in 1D, 2D, and 3D, respectively. Methods are described in Sections 3 and 4, and their basic properties are summarized in Section 5 (Tables 3, 4, and 5). Note that CD stands for contact discontinuity.

References	Dim.	Method	Comments¹²
Eulderink (1993) [83 ]	1D	Roe–Eulderink	Correct $ρshell$ with 500 zones;
			4 zones in CD.

Schneider et al. (1993) [257 ]	1D	RHLLE	$ρshell$ underestimated by 10%
			with 200 zones.

Martí and Müller (1996) [181 ]	1D	rPPM	Correct $ρshell$ with 400 zones;
			6 zones in CD.

Martí et al. (1997) [183 ]	1D, 2D	MFF-ppm	Correct $ρshell$ with 400 zones;
			6 zones in CD.

Wen et al. (1997) [295 ]	1D	rGlimm	No diffussion at discontinuities.

Yang et al. (1997) [303 ]	1D	rBS	Stable profiles.

Donat et al. (1998) [75 ]	1D	MFF-eno	Correct $ρshell$ with 400 zones;
			8 zones in CD.

Aloy et al. (1999) [6 ]	3D	MFF-ppm	Correct $ρshell$ with $√ -- 100 ∕ 3$ zones;
			2 zones in CD.

Font et al. (1999) [93 ]	1D, 3D	MFF-l	Correct $ρ shell$ with 400 zones;
			12–14 zones in CD.

	1D, 3D	Roe type-l	Correct $ρshell$ with 400 zones;
			12–14 zones in CD.

	1D, 3D	Flux split	$ρshell$ overestimated by 5%;
			8 zones in CD.

Del Zanna and Bucciantini (2002)	1D	sCENO	Correct $ρ shell$ with 400 zones;
			6 zones in CD.

Anninos and Fragile (2002)	1D, 3D	NOCD	Correct $ρshell$ with 400 zones;
			14 zones in CD.

Using artificial viscosity techniques, Centrella and Wilson [50 ] were able to reproduce the analytical solution with a 7% overshoot in $vshell$ , whereas Hawley et al. [123 ] found a 16% error in the shell density. However, when implementing a consistent formulation of artificial viscosity, like in the method developed by Anninos and Fragile [10 ], it is possible to capture the constant states in a stable manner and without noticeable errors (e.g., the shell density is underestimated by less than 2%).

The results obtained with early relativistic SPH codes [172 ] 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 [150 ]. This code also leads to a large density overshoot in the shell. Much cleaner states are obtained with the methods of Chow and Monaghan (1997) [53 ] and Siegler and Riffert (1999) [262 ], both based on conservative formulations of the SPH equations. For Chow and Monaghan’s (1997) method [53 ] the spikes at the contact discontinuity disappear but at the cost of an excessive smearing. This smearing can also be observed in Muir [204 ] (see Figures 8 and 9 ), who used the general relativistic, conservative SPH formulation of Monaghan and Price [202 ], and the dissipation method of Chow and Monaghan [53 ] to simulate Problem 1 assuming a Minkowski spacetime. Generally speaking, shock profiles obtained with relativistic SPH codes are smeared out more than those computed with HRSC methods, the shocks modelled by SPH typically being covered by more than 10 zones.

Figure 8: Density distribution for the relativistic blast wave Problem 1 defined in Table 7 at t = 0.314 obtained with the code SPH-RS-gr (see Table 5) of Muir [204

] using 5500 SPH particles and a 1D version of the code. The solid lines give the exact solutions.

Figure 9: Velocity distribution for the relativistic blast wave Problem 1 defined in Table 7 at t = 0.314 obtained with the code SPH-RS-gr (see Table 5) of Muir [204

] using 5500 SPH particles and a 1D version of the code. The solid lines give the exact solutions.

Van Putten has considered a similar initial value problem with somewhat more extreme conditions ( $vshell ≈ 0.82c$ , $σshock ≈ 5.1$ ) 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; $≈$ 50 zones for the contact discontinuity).

A movie (Figure 10 ) shows the Problem 1 blast wave evolution obtained with a modern HRSC method (the relativistic PPM method introduced in Section 3.1; code rPPM provided in Section 9.4.3). 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 [84 ] or symmetric high-order discretizations (specially the third-order schemes in [71 ]) give similar results (see Table 9). The RHLLE method [257 ] underestimates the density in the shell by about 10% in a 200 zone calculation.

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Figure 10: mpg-Movie (436 KB) The evolution of the density distribution for the relativistic blast wave Problem 1 defined in Table 7. 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.

6.2.2 Problem 2

Problem 2 was first considered by Norman and Winkler [214 ]. 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 $vshell$ = 0.960 (i.e., $Wshell$ = 3.6), while the leading shock front propagates with a velocity $vshock$ = 0.986 (i.e., $Wshock$ = 6.0). The jump in density in the shell reaches a value of 10.6. Norman and Winkler [214 ] obtained very good results with an adaptive grid of 400 zones using an implicit hydrodynamics 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 [179 , 176 , 181 , 89 , 295 , 75 ]. More recently, some symmetric HRSC codes [71 , 10 ] have also considered this problem reporting results which are competitive (as in the case of the algorithms described in [71 ]) with those obtained with Riemann solver based schemes. Table 10 gives a summary of the references where this test was considered.

Table 10: Summary of references where the blast wave problem 2 (defined in Table 7) has been considered. Shock compression ratios $σ$ are evaluated for runs with 400 numerical zones and at t $≈$ 0.40, unless otherwise established. Methods are described in Sections 3 and 4, and their basic properties are summarized in Section 5 (Tables 3, 4, and 5).

References	Method	$σ∕σexact$
Norman and Winkler (1986) [214 ]	cAV-implicit	1.00

Dubal (1991) [77 ]¹³	FCT-lw	0.80

Martí et al. (1991) [179 ]	Roe type-l	0.53

Marquina et al. (1992) [176 ]	LCA-phm	0.64

Martí and Müller (1996) [181 ]	rPPM	0.68

Falle and Komissarov (1996) [89 ]	Falle–Komissarov	0.47

Wen et al. (1997) [295 ]	rGlimm	1.00

Chow and Monaghan (1997) [53 ]	SPH-RS-c	1.16¹⁴

Donat et al. (1998) [75 ]	MFF-phm	0.60

Del Zanna and Bucciantini (2002) [71 ]	sCENO	0.69

Anninos and Fragile (2002) [10 ]	cAV-mono	1.40¹⁵

	NOCD	0.67¹⁶

A movie (Figure 11 ) 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 obtains a converged solution. The method of Falle and Komissarov [89 ] 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. [295 ] are able to handle this problem with high accuracy (see Figure 12 ). At lower resolution (400 zones) the relativistic PPM method reaches only 69% of the theoretical shock compression value (54% in case of the second-order accurate upwind method of Falle and Komissarov [89 ]; 60% with the code of Donat et al. [75 ]).

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Figure 11: mpg-Movie (1453 KB) The evolution of the density distribution for the relativistic blast wave Problem 2 defined in Table 7. 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.

Figure 12: Results from [295

] for the relativistic blast wave Problems 1 (left column) and Problem 2 (right column), respectively. Relativistic Glimm’s 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 for Lax and Lax–Wendroff methods, respectively; G refers to pure Glimm’s method.

Chow and Monaghan [53 ] 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., $vshock > 1$ ).

Anninos and Fragile [10 ] have considered Problem 2 as a test case for their artificial-viscosity based, explicit codes. They find a 40% overshoot in the shock density contrast. This demonstrates that the extra coupling introduced in the equations when using a consistent formulation of the artificial viscosity requires the usage of implicit algorithms.

6.2.3 Collision of two relativistic blast waves

The collision of two strong blast waves was used by Woodward and Colella [300 ] to compare the performance of several numerical methods in classical hydrodynamics. In the relativistic case, Yang et al. [303 ] considered this problem to test the high-order extensions of the relativistic beam scheme, whereas Martí and Müller [181 ] 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 and Müller in [181 ].

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 11. 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 Figure 13 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 $Δx$ = 0.008 (left shell) and $Δx$ = 0.019 (right shell), respectively, i.e., the entire interaction takes place in a very thin region (about 10 times smaller than in the Newtonian case where $Δx ≈ 0.2$ ).

Table 11: Initial data (pressure $p$ , density $ρ$ , velocity $v$ ) for the two relativistic blast wave collision test problem. The decay of the initial discontinuities (at x = 0.1 and x = 0.9) produces two shock waves (velocitis $vshock$ , compression ratios $σshock$ ) moving in opposite directions followed by two trailing dense shells (velocities $vshell$ , time-dependent widths $wshell$ ). The gas is assumed to be ideal with an adiabatic index $γ$ = 1.4.

	Left		Middle		Right
$p$	1000.00		0.01		100.00
$ρ$	1.00		1.0		1.00
$v$	0.00		0.00		0.00
$v shell$		0.957		–0.882
$wshell$		0.021 t		0.045 t
$vshock$		0.978		–0.927
$σshock$		14.39		9.72

Figure 13: 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 Figure 13 ) 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 $γ$ = 1.4). 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 [181 ].

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Figure 14: mpg-Movie (2049 KB) 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.

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Figure 15: mpg-Movie (698 KB) 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.

A movie (Figure 14 ) 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 and Müller [181 ]. The presence of very narrow structures involving 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).

A movie (Figure 15 ) shows the numerical solution after the interaction has occurred. Compared to the other movie (Figure 14 ), 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.