Robust Discontinuous Galerkin Methods Maintaining Physical Constraints for General Relativistic Hydrodynamics

TL;DR

This paper develops high-order discontinuous Galerkin schemes for general relativistic hydrodynamics that preserve physical constraints, eliminate spurious oscillations, and are robust in extreme conditions, advancing accurate and stable simulations.

Contribution

It introduces novel PCP-OEDG schemes with efficient oscillation elimination and robust primitive variable recovery for GRHD, ensuring physical constraints are maintained in complex scenarios.

Findings

Successfully preserves physical constraints in simulations

Handles strong shocks and high Lorentz factors effectively

Demonstrates robustness and accuracy in extreme gravitational fields

Abstract

Simulating general relativistic hydrodynamics (GRHD) presents challenges such as handling curved spacetime, achieving high-order shock-capturing accuracy, and preserving key physical constraints (positive density, pressure, and subluminal velocity) under nonlinear coupling. This paper introduces high-order, physical-constraint-preserving, oscillation-eliminating discontinuous Galerkin (PCP-OEDG) schemes with Harten-Lax-van Leer flux for GRHD. To suppress spurious oscillations near discontinuities, we incorporate a computationally efficient oscillation-eliminating (OE) procedure based on a linear damping equation, maintaining accuracy and avoiding complex characteristic decomposition. To enhance stability and robustness, we construct PCP schemes using the W-form of GRHD equations with Cholesky decomposition of the spatial metric, addressing the non-equivalence of admissible state sets in curved spacetime. We rigorously prove the PCP property of cell averages via technical estimates and the Geometric Quasi-Linearization (GQL) approach, which transforms nonlinear constraints into linear forms. Additionally, we present provably convergent PCP iterative algorithms for robust recovery of primitive variables, ensuring physical constraints are satisfied throughout. The PCP-OEDG method is validated through extensive tests, demonstrating its robustness, accuracy, and capability to handle extreme GRHD scenarios involving strong shocks, high Lorentz factors, and intense gravitational fields.