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.