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4 Numerical Schemes

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., [218Jump To The Next Citation Point219Jump To The Next Citation Point398Jump To The Next Citation Point177]). 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 [240Jump To The Next Citation Point]. Alternative techniques to finite differences, such as smoothed particle hydrodynamics, (pseudo-) spectral methods and others, are briefly considered last.

 4.1 Finite difference schemes
  4.1.1 Artificial viscosity approach
  4.1.2 High-resolution shock-capturing (HRSC) upwind schemes
  4.1.3 High-order central schemes
  4.1.4 Source terms
 4.2 Other techniques
  4.2.1 Smoothed particle hydrodynamics
  4.2.2 Spectral methods
  4.2.3 Going further
 4.3 The magnetic-field divergence-free constraint
 4.4 State-of-the-art three-dimensional codes
  4.4.1 Hydrodynamics
  4.4.2 Magnetohydrodynamics

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