GQL-Based Physical-Constraint-Preserving High-Order Finite Difference Schemes for Special Relativistic Hydrodynamics in Arbitrary Dimensions
This work provides a robust and efficient method to prevent numerical breakdown in high-order simulations of relativistic hydrodynamics, which is critical for astrophysical applications.
The authors developed a physical-constraint-preserving flux-limiting framework for high-order finite difference schemes in special relativistic hydrodynamics, using geometric quasilinearization to enforce positivity and subluminal velocity. The method achieves robust enforcement of physical constraints, sharp discontinuity resolution, and design-order accuracy in 1D to 3D tests.
High-order accurate simulations of special relativistic hydrodynamics (RHD) are prone to numerical breakdown if intrinsic physical constraints (positive rest-mass density/pressure and subluminal velocity) are violated near strong discontinuities. In this work, we develop a robust and efficient physical-constraint-preserving (PCP) flux-limiting framework for high-order schemes, using finite-difference WENO as a representative example. By leveraging the geometric quasilinearization (GQL) representation, which equivalently reformulates the nonlinear RHD constraints into a family of linear inequalities, we integrate a Zalesak-type Flux-Corrected Transport (FCT) update into a scalar-style limiter that acts directly on conservative variables. A critical innovation is the explicit, non-iterative determination of limiting parameters via a rational stereographic parameterization of the GQL normal vector. This technique transforms the required worst-case minimization over auxiliary variables into a generalized Rayleigh-quotient formulation, allowing the optimal parameters to be obtained by solving small symmetric eigenvalue problems ($2\times2$ in 1D; $(d+1)\times(d+1)$ in $d$ dimensions). Relaxed variants are further introduced to reduce computational costs in multidimensions while retaining the PCP guarantee. Extensive numerical benchmarks ranging from 1D to 3D, including ultra-relativistic Riemann problems and astrophysical jets, demonstrate that the proposed method robustly enforces physical admissibility, sharply resolves discontinuities, and maintains design-order accuracy for smooth solutions.