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 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 6 contains a
summary of the results obtained for the shock heating test by
various authors.
Explicit finite difference techniques based on a
non-conservative formulation of the hydrodynamic equations and on
non-consistent artificial viscosity [50, 123
, 10
] (or even consistent
artificial viscosity [10
]) are able to handle
flow Lorentz factors up to
with moderately large errors
(
) at best [298, 187
]. Norman and
Winkler [213
] 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 [179, 176
, 83
, 256
, 84
, 181
, 89
]. More recently,
HRSC methods based on symmetric discretizations [71
, 10
] have also
demonstrated the same capability to describe highly relativistic
flows. 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 [74
, 295
, 6
, 10
].
Schneider et al. [256] have compared the
accuracy of a code based on the RHLLE Riemann solver with different
versions of relativistic FCT codes for inflow Lorentz factors in
the range 1.5 to 50. They find that the error in
is reduced by a factor of two when using HLL. Further tests of the
(1D) RHLLE method were performed by Rischke et al. [244
, 246
, 245] who considered expansion
into vacuum, semi-infinite colliding slabs, and spherically and
cylindrically symmetric expansions for equations of state for both
thermodynamically normal and anomalous matter (see Section 7.3). In the latter two test cases
RHLLE transport is done in the radial direction while corrections
due to geometry are implemented via Sod’s method. Rischke et
al. [244
, 246
] also present a
detailed comparison of the RHLLE method and relativistic
extensions [113] of flux-corrected
transport (FCT) algorithms [33
, 35, 34]. They find that not all
versions of the numerical algorithms explored in their
investigation can be straightforwardly applied. Moreover, numerical
parameters like the grid spacing or the antidiffusion coefficients
(for FCT SHASTA) must be chosen with care, in order to produce
solutions which are free of numerical artifacts. Studying the
“slab-on-slab” collision test problem (up to flow Lorentz factors
of 2.3) they particularly find [246
] that analytical
solutions are reproduced remarkably well with RHLLE and also with
FCT SHASTA, provided the numerical diffusion is sufficiently large
(i.e., when the antidiffusion in SHASTA is chosen sufficiently
small).
Within SPH methods, Chow and Monaghan [53] 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 and Riffert [261
] 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 [261
]; 20-24 in the code
of [53
]) compared with the
typical dissipation of HRSC methods (see below).
|
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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
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
is smaller than
, 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.
The quality of the results obtained with high-order symmetric
schemes [10
, 71
] is similar.
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 [249, 183
, 295
]. Aloy et
al. [6
] 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 error for the density and internal
energy in a few cells near the reflecting wall. This ’overheating’,
as it is known in classical hydrodynamics [212
], is a numerical
artifact which is considerably reduced when Marquina’s scheme is
used [76]. In passing we note that
the strong overheating found by Noh [212
] for the spherical
shock reflection test using PPM (Figure 24 in [212]) 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. The piece-wise linear method described
in [249
] gives an undershoot
of 14% in case of ultra-relativistic flows (e.g., Table 1 and
Figure 1 in [249
]).