Numerical Relativistic Hydrodynamics: HRSC Methods

AbstractThis chapter is devoted to the analysis of those numerical methods based on the conservative formulation of the equations, as is the case of the relativistic-hydrodynamics equation. Conservative schemes are discussed both in their finite-volume version and in their finite-difference version, which differ in the fact of evolving in time the cell-averaged or the point-values of the solution, respectively. Upwind methods are then introduced, showing how the solution of the Riemann problem can be exploited to incorporate in a natural way the upwind property even in nonlinear systems of equations. Total variation diminishing methods are then discussed, which allow for larger than first-order accuracy and a selected class of Riemann solvers is presented, of most practical use in relativistic hydrodynamics. The integration in time is analysed through Runge–Kutta methods, including the case of stiff source terms. A brief but clear exposition of central scheme closes the chapter.