As HRSC methods are written in conservation form, the time evolution of zone averaged state vectors is governed by some functions (the numerical fluxes) evaluated at zone interfaces. Numerical fluxes are mostly obtained by means of an exact or approximate Riemann solver. High resolution is usually achieved by using monotonic polynomials in order to interpolate the approximate solutions within numerical cells.

Solving Riemann problems exactly involves time-consuming computations, which are particularly costly in the case of multidimensional SRHD due to the coupling of the equations through the Lorentz factor (see Section 2.3). Therefore, as an alternative, the usage of approximate Riemann solvers has been proposed.

In this section we summarize how the numerical fluxes are computed in a number of methods for numerical SRHD. Methods based on exact Riemann solvers are discussed in Sections 3.1 and 3.2, while those based on approximate solvers are discussed in Sections 3.3, 3.4, 3.5, 3.6, and 3.7 . Readers not familiar with HRSC methods are referred to Section 9.4, where the basic properties of these methods are described and an outline of the recent developments is given.

- 3.1 Relativistic PPM
- 3.2 The relativistic Glimm method
- 3.3 Two-shock approximation for relativistic hydrodynamics
- 3.4 Roe-type relativistic solvers
- 3.5 Falle and Komissarov upwind scheme
- 3.6 Relativistic HLL method
- 3.7 Marquina's flux formula
- 3.8 Symmetric TVD schemes with nonlinear numerical dissipation

Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
http://www.livingreviews.org/lrr-1999-3
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