3 High-Resolution Shock-Capturing Methods

The application of high-resolution shock-capturing (HRSC) methods caused a revolution in numerical SRHD. These methods satisfy in a quite natural way the basic properties required for any acceptable numerical method:

• high order of accuracy,
• stable and sharp description of discontinuities, and
• convergence to the physically correct solution.

Moreover, HRSC methods are conservative, and because of their shock capturing property discontinuous solutions are treated both consistently and automatically whenever and wherever they appear in the flow.

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, although symmetric schemes can also be implemented. 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 multi-dimensional 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 remainder of this section we summarize the computation of the numerical fluxes 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, 3.7, and 3.8. Symmetric schemes are also presented in Section 3.9. Readers not familiar with HRSC methods are referred to Section 9.5, where the basic properties of these methods as well as an outline of the recent developments are described. Let us note that the focus of our review are one-dimensional versions of the numerical methods and algorithms. Multi-dimensional flow problems can be handled by standard means which are briefly reviewed in Section 9.5.