A Piecewise-Parabolic Reconstruction of the Physical Variables in a Low-Dissipation HLL Method for the Numerical Solution of the Equations of Special Relativistic Hydrodynamics

A construction of the original HLL method for solving problems of relativistic hydrodynamics by using a piecewise-parabolic reconstruction of the physical variables is described. The resulting numerical method makes it possible to reproduce the numerical solutions with small dissipation at the discontinuities. The method is verified in problems of discontinuity breakdown in one-dimensional and two-dimensional formulation. The accuracy of the numerical scheme is studied in one-dimensional discontinuity breakdown problems. The method is also tested in typical astrophysical problems: interaction of relativistic jets, collision of clouds at relativistic speeds, and supernova explosion.