Using low dissipation Lax-Friedrichs scheme for numerical modeling of relativistic flows

The Lax-Friedrichs scheme is traditionally considered an alternative to the Godunov scheme, since it does not require solving the Riemann problem. In the equations of special relativistic hydrodynamics, the speed of light is a natural limitation of the wave propagation speed. The use of such an upper estimate of the slopes of characteristics in the schemes of Roe, the Rusanov type, or the Harten-Lax-Van Leer family leads to a construction equivalent to the Lax-Friedrichs scheme. Due to the absolute robustness of the scheme, a number of software implementations have been developed on its basis for modeling relativistic gas flows. In this paper, we propose a piecewise parabolic reconstruction of the physical variables to reduce dissipation of the numerical method. The use of such a reconstruction in the Lax-Friedrichs scheme allows us to obtain an absolutely robust simple scheme of high-order accuracy on smooth solutions and with small dissipation at the discontinuities. The computational experiments carried out in the article confirm these properties of the scheme.