

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.
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 , density
, velocity ) for two common relativistic
blast wave test problems. The decay of
the initial discontinuity leads to a shock wave (velocity
, compression ratio ) and the formation of a dense shell
(velocity , time-dependent
width ) both
propagating to the right. The gas is assumed
to be ideal with an adiabatic index . |
|
|
|
|
|
|
|
|
|
Problem 1 |
|
|
Problem 2 |
|
|
Left |
|
Right |
Left |
|
Right |
|
|
|
|
|
|
|
 |
 |
|
 |
 |
|
 |
 |
 |
|
 |
 |
|
 |
 |
 |
|
 |
 |
|
 |
|
|
|
|
|
|
|
 |
|
 |
|
|
 |
|
 |
|
 |
|
|
 |
|
 |
|
 |
|
|
 |
|
 |
|
 |
|
|
 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
6.2.1
Problem 1
In Problem 1, the decay of the initial
discontinuity gives rise to a dense shell of matter with velocity
(
)
propagating to the right. The shell trailing a shock wave of speed
increases its width
according to
, i.e.,
at time
the shell covers about 4% of the grid
(
). Tables 8 and 9 give a
summary of the references where this test was considered for
non-HRSC and HRSC methods, respectively.
|
|
|
|
|
|
|
|
|
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; |
|
|
|
overestimated
by 16%. |
|
|
|
|
Dubal (1991) [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; |
|
|
|
overestimated by 5%. |
|
|
|
|
van Putten (1993) [287 ] |
1D |
van Putten |
Stable profiles; |
|
|
|
excessive smearing, especially of
the |
|
|
|
CD ( zones). |
|
|
|
|
Schneider et al. (1993) [256 ] |
1D |
SHASTA-c |
Non-monotonic intermediate states; |
|
|
|
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) [261 ] |
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 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 |
|
|
|
|
|
|
|
|
|
|
|
|
|
Eulderink (1993) [83 ] |
1D |
Roe-Eulderink |
Correct with 500
zones; |
|
|
|
4 zones in CD. |
|
|
|
|
Schneider et al. (1993) [256 ] |
1D |
RHLLE |
underestimated by 10% |
|
|
|
with 200 zones. |
|
|
|
|
Martí and Müller (1996) [181 ] |
1D |
rPPM |
Correct with 400
zones; |
|
|
|
6 zones in CD. |
|
|
|
|
Martí et al. (1997) [183 ] |
1D, 2D |
MFF-ppm |
Correct 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 with 400
zones; |
|
|
|
8 zones in CD. |
|
|
|
|
Aloy et al. (1999) [6 ] |
3D |
MFF-ppm |
Correct with zones; |
|
|
|
2 zones in CD. |
|
|
|
|
Font et al. (1999) [93 ] |
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. |
|
|
|
|
Del Zanna and Bucciantini (2002) |
1D |
sCENO |
Correct with 400
zones; |
|
|
|
6 zones in CD. |
|
|
|
|
Anninos and Fragile (2002) |
1D, 3D |
NOCD |
Correct with 400
zones; |
|
|
|
14 zones in CD. |
|
|
|
|
|
|
|
|
|
|
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. |
|
Using artificial viscosity techniques, Centrella
and Wilson [50] were able to reproduce
the analytical solution with a 7% overshoot in
, 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) [261
], 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.
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 (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 [256
] underestimates the
density in the shell by about 10% in a 200 zone calculation.
6.2.2
Problem 2
Problem 2 was first considered by Norman
and Winkler[213
]. 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
. The fluid in the shell moves with
(i.e.,
), while the leading shock
front propagates with a velocity
(i.e.,
). The jump in density in the shell
reaches a value of 10.6. Norman and Winkler [213
] 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.
|
|
|
|
|
|
|
|
References |
Method |
 |
|
|
|
|
|
|
|
|
|
|
|
Norman and Winkler (1986) [213 ] |
cAV-implicit |
1.00 |
|
|
|
Dubal (1991) [77 ] |
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 |
1.16 |
|
|
|
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 |
1.40 |
|
|
|
|
NOCD |
0.67 |
|
|
|
|
|
|
|
|
Table 10: |
Summary of references
where the blast wave problem 2
(defined in Table 7) has been considered.
Shock compression ratios are
evaluated for runs with 400 numerical zones and at , 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). |
|
An MPEG 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
]).
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.,
).
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
and
), 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
for the shock wave propagating to the right and
left, respectively). Hence, the shock interaction occurs later (at
) than in the Newtonian problem (at
). The top panel in Figure 13 shows four snapshots
of the density distribution including the moment of the collision
of the blast waves at
and
. At the time of collision the two shells have a
width of
(left shell) and
(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
).
Table 11: |
Initial data
(pressure , density
, velocity ) for the two relativistic blast
wave collision test problem. The decay
of the initial discontinuities (at and ) produces two shock
waves (velocitis , compression ratios
) moving in opposite directions
followed by two trailing dense shells
(velocities ,
time-dependent widths ). The gas is assumed to be ideal with an adiabatic index . |
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
). The solution just described applies
until
, when the next interaction takes place.
The complete analytical solution before and after
the collision up to time
can be obtained following
Appendix II in [181
].
An MPEG movie (Figure 14
) shows the evolution of the density up to the
time of shock collision at
. 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
. The errors
drop only slightly to about 40% (5%) at the time of collision (
).
An MPEG movie (Figure 15
) shows the numerical solution
after the interaction has occurred. Compared to the other MPEG
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.

