We turn now to describing the numerical schemes, mainly those based on finite differences, specifically
designed to solve nonlinear hyperbolic systems of conservation laws. As discussed in the previous two
sections, both the equations of general-relativistic hydrodynamics and magnetohydrodynamics fall in this
category, irrespective of the formulation employed. Even though we also consider schemes based on artificial
viscosity techniques, the emphasis is on the 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., [218, 219
, 398
, 177]).
For this reason only the most relevant features will be covered here, directing 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 [240
]. Alternative techniques to finite differences, such as
smoothed particle hydrodynamics, (pseudo-) spectral methods and others, are briefly considered
last.
http://www.livingreviews.org/lrr-2008-7 | ![]() This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |