Problem 1 was a demanding problem for relativistic
hydrodynamic codes in the mid eighties [28,
75
], 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).
Using artificial viscosity techniques, Centrella &
Wilson [28] were able to reproduce the analytical solution with a 7%
overshoot in
, whereas Hawley et al. [75
] got a 16% error in the shell density.
The results obtained with early relativistic SPH codes [102] 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 [89
]. 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) [30
] and Siegler & Riffert (1999) [164
], both based on conservative formulations of the SPH equations.
In the case of Chow & Monaghan's (1997) method [30
], 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 (,
) 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 (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 [50] give similar results (see Table
7). The relativistic HLL method [161
] underestimates the density in the shell by about 10% in a 200
zone calculation.
References | Dim. | Method | Comments |
Centrella & Wilson
(1984) [28] |
1D | AV-mono | Stable profiles without oscillations. Velocity overestimated by 7%. |
Hawley et al.
(1984) [75 ![]() |
1D | AV-mono | Stable profiles without oscillations.
![]() |
Dubal
(1991) [45 ![]() |
1D | FCT-lw | 10-12 zones at the CD. Velocity overestimated by 4.5%. |
Mann
(1991) [102 ![]() |
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.
![]() |
van Putten
(1993) [181 ![]() |
1D | van Putten | Stable profiles. Excessive smearing, specially at the CD
(![]() |
Schneider et al.
(1993) [161 ![]() |
1D | SHASTA-c | Non monotonic intermediate states.
![]() |
Chow & Monaghan
(1997) [30 ![]() |
1D | SPH-RS-c | Stable profiles without spikes. Excessive smearing at the CD and at the shock. |
Siegler & Riffert
(1999) [164 ![]() |
1D | SPH-cAV-c | Correct constant states. Large amplitude spikes at the
CD. Excessive smearing at the shock transition (![]() |
References | Dim. | Method |
Comments
![]() |
Eulderink
(1993) [49 ![]() |
1D | Roe-Eulderink | Correct
![]() |
Schneider et al.
(1993) [161 ![]() |
1D | HLL-l |
![]() |
Martí & Müller
(1996) [109 ![]() |
1D | rPPM | Correct
![]() |
Martí et al.
(1997) [111 ![]() |
1D, 2D | MFF-ppm | Correct
![]() |
Wen et al.
(1997) [187 ![]() |
1D | rGlimm | No diffusion at discontinuities. |
Yang et al.
(1997) [194 ![]() |
1D | rBS | Stable profiles. |
Donat et al.
(1998) [43 ![]() |
1D | MFF-eno | Correct
![]() |
Aloy et al.
(1999) [3 ![]() |
3D | MFF-ppm | Correct
![]() ![]() |
Font et al.
(1999) [59 ![]() |
1D, 3D | MFF-l | Correct
![]() |
1D, 3D | Roe type-l | Correct
![]() |
|
1D, 3D | Flux split |
![]() |
Several HRSC methods based on relativistic Riemann solvers
have used Problem 2 as a standard test [107,
106
,
109
,
55
,
187
,
43
]. Table
8
gives a summary of the references where this test was
considered.
References | Method |
![]() |
Norman & Winkler (1986) [131![]() |
cAV-implicit | 1.00 |
Dubal (1991)
![]() ![]() |
FCT-lw | 0.80 |
Martí et al. (1991) [107![]() |
Roe type-l | 0.53 |
Marquina et al. (1992) [106] | LCA-phm | 0.64 |
Martí & Müller (1996) [109![]() |
rPPM | 0.68 |
Falle & Komissarov (1996) [55![]() |
Falle-Komissarov | 0.47 |
Wen et al. (1997) [187![]() |
rGlimm | 1.00 |
Chow & Monaghan (1997) [30![]() |
SPH-RS-c | 1.16
![]() |
Donat et al. (1998) [43![]() |
MFF-phm | 0.60 |
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 [55] 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. [187
] 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. [43
]).
Chow & Monaghan [30] 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.,
).
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
(left shell) and
(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
).
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
). 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 [109].
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 [109]. 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.
![]() |
Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller http://www.livingreviews.org/lrr-1999-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |