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.