In the mid eighties, Norman & Winkler [131] proposed a reformulation of t he difference equations of SRHD
with an artificial viscosity consistent with the relativistic
dynamics of non-perfect fluids. The strong coupling introduced in
the equations by the presence of the viscous terms in the
definition of relativistic momentum and total energy densities
required an implicit treatment of the difference equations.
Accurate results across strong relativistic shocks with large
Lorentz factors were obtained in combination with adaptive mesh
techniques. However, no multidimensional version of this code was
developed.
Attempts to integrate the RHD equations avoiding the use of
artificial viscosity were performed in the early nineties.
Dubal [45] developed a 2D code for relativistic magneto-hydrodynamics
based on an explicit second-order Lax-Wendroff scheme
incorporating a flux corrected transport (FCT) algorithm [20
]. Following a completely different approach Mann [102
] proposed a multidimensional code for general relativistic
hydrodynamics based on smoothed particle hydrodynamics (SPH)
techniques [121
], which he applied to relativistic spherical collapse [104
]. When tested against 1D relativistic shock tubes all these
codes performed similar to the code of Wilson. More recently,
Dean et al. [39
] have applied flux correcting algorithms for the SRHD equations
in the context of heavy ion collisions. Recent developments in
relativistic SPH methods [30
,
164
] are discussed in Section
4.2
.
A major break-through in the simulation of ultra-relativistic
flows was accomplished when high-resolution shock-capturing
(HRSC) methods, specially designed to solve hyperbolic systems of
conservations laws, were applied to solve the SRHD equations [107,
106
,
49
,
50
]. This review is intended to provide a comprehensive discussion
of different HRSC methods and of related methods used in SRHD.
Numerical methods for special relativistic MHD flows (MHD stands
for magneto hydrodynamics) are not included, because they are
beyond the scope of this review. However, we may include such a
discussion in a future update of this article.
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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 |