In the mid eighties, Norman & Winkler  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  developed a 2D code for relativistic magneto-hydrodynamics based on an explicit second-order Lax-Wendroff scheme incorporating a flux corrected transport (FCT) algorithm . Following a completely different approach Mann  proposed a multidimensional code for general relativistic hydrodynamics based on smoothed particle hydrodynamics (SPH) techniques , which he applied to relativistic spherical collapse . When tested against 1D relativistic shock tubes all these codes performed similar to the code of Wilson. More recently, Dean et al.  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.
|Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
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