High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics

TL;DR

This paper introduces high-order finite difference WENO schemes that preserve physical constraints for special relativistic hydrodynamics, ensuring accuracy and robustness in challenging scenarios like high Lorentz factors and strong discontinuities.

Contribution

It extends positivity-preserving WENO schemes to relativistic hydrodynamics by developing new methods to handle the system's complex coupling and physical constraints.

Findings

Demonstrates high accuracy in numerical tests

Shows robustness with large Lorentz factors

Effective in problems with strong discontinuities

Abstract

The paper develops high-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamical (RHD) equations, built on the local Lax-Friedrich splitting, the WENO reconstruction, the physical-constraints-preserving flux limiter, and the high-order strong stability preserving time discretization. They are extensions of the positivity-preserving finite difference WENO schemes for the non-relativistic Euler equations. However, developing physical-constraints-preserving methods for the RHD system becomes much more difficult than the non-relativistic case because of the strongly coupling between the RHD equations, no explicit expressions of the primitive variables and the flux vectors, in terms of the conservative vector, and one more physical constraint for the fluid velocity in addition to the positivity of the rest-mass density and the pressure. The key is to prove the convexity and other properties of the admissible state set and discover a concave function with respect to the conservative vector replacing the pressure which is an important ingredient to enforce the positivity-preserving property for the non-relativistic case. Several one- and two-dimensional numerical examples are used to demonstrate accuracy, robustness, and effectiveness of the proposed physical-constraints-preserving schemes in solving RHD problems with large Lorentz factor, or strong discontinuities, or low rest-mass density or pressure etc.