High-order accurate physical-constraints-preserving finite difference WENO schemes for special relativistic hydrodynamics
For computational astrophysics, this provides provably physical-constraints-preserving high-order schemes for relativistic flows, addressing a known bottleneck in RHD simulations.
The paper develops high-order accurate finite difference WENO schemes that preserve physical constraints (positivity of density, pressure, and subluminal velocity) for special relativistic hydrodynamics, extending positivity-preserving methods from non-relativistic Euler equations. Numerical tests demonstrate accuracy and robustness for problems with large Lorentz factors or strong discontinuities.
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.