Method | Ultra-relativistic regime |
Handling of discontinuities
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Extension to several spatial dimensions
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Extension to
GRHD RMHD |
|
AV-mono |
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O, SE |
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cAV-implicit |
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HRSC
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rGlimm |
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sTVD |
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D |
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van Putten |
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D |
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FCT |
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O |
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SPH |
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D, O |
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Since their introduction in numerical RHD at the beginning of nineties, HRSC methods have demonstrated their ability to describe accurately (stable and without excessive smearing) relativistic flows of arbitrarily large Lorentz factors and strong discontinuities, reaching the same quality as in classical hydrodynamics. In addition (as it is the case for classical flows, too), HRSC methods show the best performance compared to any other method (e.g., artificial viscosity, FCT or SPH).
Despite of the latter fact, a lot of effort has been put into
improving these non-HRSC methods. Using a consistent formulation
of artificial viscosity has significantly enhanced the capability
of finite difference schemes [131] as well as of relativistic SPH [164
] to handle strong shocks without spurious post-shock
oscillations. However, this comes at the price of a large
numerical dissipation at shocks. Concerning relativistic SPH,
recent investigations using a conservative formulation of the
hydrodynamic equations [30
,
164] have reached an unprecedented accuracy with respect to previous
simulations, although some issues still remain. Besides the
strong smearing of shocks, the description of contact
discontinuities and of thin structures moving at
ultra-relativistic speeds needs to be improved (see Section
6.2).
Concerning FCT techniques, those codes based on a conservative
formulation of the RHD equations have been able to handle
relativistic flows with discontinuities at all flow speeds,
although the quality of the results is below that of HRSC methods
in all cases [161].
The extension to multi-dimensions is simple for most
relativistic codes. Finite difference techniques are easily
extended using directional splitting. Note, however, that HRSC
methods based on exact solutions of the Riemann problem [109,
187
] first require the development of a multidimensional version of
the relativistic Riemann solver. The adapting-grid, artificial
viscosity, implicit code of Norman & Winkler [131] and the relativistic Glimm method of Wen et al. [187] are restricted to one dimensional flows. Note that Glimm's
method produces the best results in all the tests analyzed in
Section
6
.
The symmetric TVD scheme proposed by Davis [38] and extended to GRMHD (see below) by Koide et al. [82
] combines several characteristics making it very attractive. It
is written in conservation form and is TVD, i.e., it is
converging to the physical solution. In addition, it is
independent of spectral decompositions, which allows for a simple
extension to RMHD. Quite similar statements can be made about the
approach proposed by van Putten [181
]. In contrast to FCT schemes (which are also easily extended to
general systems of equations), both Koide et al.'s and van
Putten's methods are very stable when simulating mildly
relativistic flows (maximum Lorentz factors
) with discontinuities. Their only drawback is an excessive
smearing of the latter. A comparison of Davis' method with
Riemann solver based methods would be desirable.
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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 |