3 Numerical Schemes 

We turn now to describe the numerical schemes, mainly those based on finite differences, specifically designed to solve nonlinear hyperbolic systems of conservation laws. As discussed in the previous section, the equations of general relativistic hydrodynamics fall in this category, irrespective of the formulation. Even though we also consider schemes based on artificial viscosity techniques, the emphasis is 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 structure 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., [152, 153, 287, 128]). For this reason only the most relevant features will be covered here, addressing the reader to the appropriate literature for further details. In particular, an excellent introduction to the implementation of HRSC schemes in special relativistic hydrodynamics is presented in the Living Reviews article by Martí and Müller [164]. Alternative techniques to finite differences, such as smoothed particle hydrodynamics, (pseudo-)spectral methods and others, are briefly considered last.

• 3.1 Finite difference schemes
• 3.2 Other techniques
• 3.3 State-of-the-art three-dimensional codes

Numerical Hydrodynamics in General Relativity José A. Font http://www.livingreviews.org/lrr-2003-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de