3.1 Relativistic PPMNumerical Hydrodynamics in Special Relativity2.3 Exact solution of the

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: (i) high order of accuracy, (ii) stable and sharp description of discontinuities, and (iii) 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. 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 PPMNumerical Hydrodynamics in Special Relativity2.3 Exact solution of the

image Numerical Hydrodynamics in Special Relativity
Jose Maria Martí and Ewald Müller
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de