1.3 Plan of the review1 Introduction1.1 Current fields of research

1.2 Overview of the numerical methods 

The first attempt to solve the equations of relativistic hydrodynamics (RHD) was made by Wilson [188, 189] and collaborators [28Jump To The Next Citation Point In The Article, 75Jump To The Next Citation Point In The Article] using an Eulerian explicit finite difference code with monotonic transport. The code relies on artificial viscosity techniques [185, 154Jump To The Next Citation Point In The Article] to handle shock waves. It has been widely used to simulate flows encountered in cosmology, axisymmetric relativistic stellar collapse, accretion onto compact objects and, more recently, collisions of heavy ions. Almost all the codes for numerical both special (SRHD) and general (GRHD) relativistic hydrodynamics developed in the eighties [142, 167, 126, 125, 127, 51] were based on Wilson's procedure. However, despite its popularity it turned out to be unable to describe extremely relativistic flows (Lorentz factors larger than 2; see, e.g., [28Jump To The Next Citation Point In The Article]) accurately.

In the mid eighties, Norman & Winkler [131Jump To The Next Citation Point In The Article] 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 [45Jump To The Next Citation Point In The Article] developed a 2D code for relativistic magneto-hydrodynamics based on an explicit second-order Lax-Wendroff scheme incorporating a flux corrected transport (FCT) algorithm [20Jump To The Next Citation Point In The Article]. Following a completely different approach Mann [102Jump To The Next Citation Point In The Article] proposed a multidimensional code for general relativistic hydrodynamics based on smoothed particle hydrodynamics (SPH) techniques [121Jump To The Next Citation Point In The Article], which he applied to relativistic spherical collapse [104Jump To The Next Citation Point In The Article]. When tested against 1D relativistic shock tubes all these codes performed similar to the code of Wilson. More recently, Dean et al. [39Jump To The Next Citation Point In The Article] have applied flux correcting algorithms for the SRHD equations in the context of heavy ion collisions. Recent developments in relativistic SPH methods [30Jump To The Next Citation Point In The Article, 164Jump To The Next Citation Point In The Article] 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 [107Jump To The Next Citation Point In The Article, 106Jump To The Next Citation Point In The Article, 49Jump To The Next Citation Point In The Article, 50Jump To The Next Citation Point In The Article]. 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.



1.3 Plan of the review1 Introduction1.1 Current fields of research

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
© Max-Planck-Gesellschaft. ISSN 1433-8351
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