3 Numerical Schemes 

This section describes the numerical schemes, mainly those based on finite differences, specifically designed to solve non-linear hyperbolic systems of conservation laws. As discussed in the previous section, the equations of general relativistic hydrodynamics fall in this category. Although schemes based on artificial viscosity techniques are also considered, the emphasis is given on the so-called high-resolution shock-capturing (HRSC) schemes (or Godunov-type methods), based on (either exact or approximate) solutions of local Riemann problems using the characteristic fields of the equations. Such finite difference schemes (or, in general, finite volume schemes) have been the subject of diverse review articles and textbooks (see, e.g., [117, 118, 100]). For this reason only the most relevant features will be covered here, referring the reader to the appropriate literature for further details. In particular, an excellent introduction on the implementation of HRSC schemes in special relativistic hydrodynamics is presented in the Living Reviews article by Martí and Müller [126]. Alternative techniques to finite differences, such as Smoothed Particle Hydrodynamics and (pseudo-) Spectral Methods, are briefly considered last.

• 3.1 Finite difference schemes
• 3.2 Other techniques

Numerical Hydrodynamics in General Relativity José A. Font http://www.livingreviews.org/lrr-2000-2 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de