1.3 Plan of the review

This review is intended to provide a comprehensive discussion of different HRSC methods and of related methods used in SRHD. However, we are not going to consider finite difference and finite volume methods based on the usage of artificial viscosity techniques which are reviewed, e.g., in the book of Wilson and Mathews [298]. Numerical methods for special relativistic MHD flows are also not included as they are beyond the scope of this review. Furthermore, we do not include numerical methods for general relativistic hydrodynamics. A comprehensive and recent discussion of such methods can be found in another article in Living Reviews in Relativity written by Font [91Jump To The Next Citation Point].

The review is organized as follows. Section 2 contains a derivation of the equations of special relativistic (perfect) fluid dynamics, as well as a discussion of their main properties. In Section 3 the most recent developments in numerical methods for SRHD are reviewed paying particular attention to high-resolution shock-capturing methods.

We have focussed on those aspects of the numerical methods more specific of SRHD, i.e., the discussion of relativistic Riemann solvers and the computation of numerical fluxes. Some comments about the extension to multi-dimensional flows are included in Section 9 (see below).

Other developments in special relativistic numerical hydrodynamics are discussed in Section 4. Numerical results obtained with different methods as well as analytical solutions for several test problems are presented in Section 6. Two astrophysical applications of SRHD are discussed in Section 7. An evaluation of the various numerical methods is given in Section 8 together with an outlook for future developments. Finally, some additional technical information including the incorporation of general equations of state is presented in Section 9.

The reader is assumed to have basic knowledge in classical [15362Jump To The Next Citation Point] and relativistic fluid dynamics [2748Jump To The Next Citation Point], as well as in finite difference/volume methods for partial differential equations [237215]. A discussion of modern finite volume methods for hyperbolic systems of conservation laws can be found, e.g., in [157Jump To The Next Citation Point160Jump To The Next Citation Point154Jump To The Next Citation Point]. The theory of spectral methods for fluid dynamics is developed in [42], and smoothed particle hydrodynamics is reviewed in [199Jump To The Next Citation Point].

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