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; [111] 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.,
) gas with coordinate velocity
and Lorentz factor
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 (
). Due to conservation of energy across the shock the gas has a
specific internal energy given by
The compression ratio of shocked and unshocked gas,
, follows from
where
is the adiabatic index of the equation of state. The shock
velocity is given by
In the unshocked region () the pressure-less gas flow is self-similar and has a density
distribution given by
where
for planar, cylindrical or spherical geometry, and where
is the density of the inflowing gas at infinity (see Fig.
3).
In the Newtonian case the compression ratio
of shocked and unshocked gas cannot exceed a value of
independently of the inflow velocity. This is different for
relativistic flows, where
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
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 |
![]() |
Method |
![]() |
![]() |
Centrella & Wilson
(1984) [28 ![]() |
0 | AV-mono | 2.29 |
![]() |
Hawley et al.
(1984) [75 ![]() |
0 | AV-mono | 4.12 |
![]() |
Norman & Winkler
(1986) [131 ![]() |
0 | cAV-implicit | 10.0 | 0.01 |
McAbee et al.
(1989) [113 ![]() |
0 | AV-mono | 10.0 | 2.6 |
Martí et al.
(1991) [107 ![]() |
0 | Roe type-l | 23 | 0.2 |
Marquina et al.
(1992) [106 ![]() |
0 | LCA-phm | 70 | 0.1 |
Eulderink
(1993) [49 ![]() |
0 | Roe-Eulderink | 625 |
![]() ![]() |
Schneider et al.
(1993) [161 ![]() |
0 | HLL-l |
![]() |
0.2
![]() |
0 | SHASTA-c |
![]() |
0.5
![]() |
|
Dolezal & Wong
(1995) [42 ![]() |
0 | LCA-eno |
![]() |
![]() ![]() |
Martí & Müller
(1996) [109 ![]() |
0 | rPPM | 224 | 0.03 |
Falle & Komissarov
(1996) [55 ![]() |
0 | Falle-Komissarov | 224 |
![]() ![]() |
Romero et al.
(1996) [157 ![]() |
2 | Roe type-l | 2236 | 2.2 |
Martí et al.
(1997) [111 ![]() |
1 | MFF-ppm | 70 | 1.0 |
Chow & Monaghan
(1997) [30 ![]() |
0 | SPH-RS-c | 70 | 0.2 |
Wen et al.
(1997) [187 ![]() |
2 | rGlimm | 224 |
![]() |
Donat et al.
(1998) [43 ![]() |
0 | MFF-eno | 224 |
![]() ![]() |
Aloy et al.
(1999) [3 ![]() |
0 | MFF-ppm |
![]() |
3.5
![]() |
Sieglert & Riffert
(1999) [164 ![]() |
0 | SPH-cAV-c | 1000 |
![]() ![]() |
Explicit finite-difference techniques based on a
non-conservative formulation of the hydrodynamic equations and on
non-consistent artificial viscosity [28,
75
] are able to handle flow Lorentz factors up to
with moderately large errors (
) at best [190,
113]. Norman & Winkler [131
] got very good results (
) 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 [107,
106
,
49
,
161
,
50
,
109
,
55
]. 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 [42
,
187
,
3
].
Schneider et al. [161] 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
was reduced by a factor of two when using HLL.
Within SPH methods, Chow & Monaghan [30] have obtained results comparable to those of HRSC methods (
) for flow Lorentz factors up to 70, using a relativistic SPH
code with Riemann solver guided dissipation. Sieglert &
Riffert [164
] 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. [164
]; 20-24 in the code of ref. [30
]) 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
(
). These results are obtained with the relativistic PPM code
of [109
], which uses an exact Riemann solver based on the procedure
described in Section
2.3
.
The shock wave is resolved by three zones and there are no
post-shock numerical oscillations. The density increases by a
factor
across the shock. Near
x
=0 the density distribution slightly undershoots the analytical
solution (by
) 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
, 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 [157,
111
,
187
]. Aloy et al. [3
] 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 [130], 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 [130
] 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 [157
]).
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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 |