Robust Discontinuous Galerkin Methods Maintaining Physical Constraints for General Relativistic Hydrodynamics

TL;DR

This work develops high-order physical-constraint-preserving oscillation-eliminating discontinuous Galerkin (PCP-OEDG) schemes for general relativistic hydrodynamics (GRHD) in arbitrary spacetimes. By formulating GRHD in the W-form and employing a Geometric Quasi-Linearization (GQL) framework, the authors establish spacetime-invariant admissible state sets and provable PCP properties for cell averages, complemented by a pointwise PCP limiter and provably convergent primitive-variable recovery algorithms. An oscillation-eliminating (OE) procedure based on an exact linear damping equation is embedded after each RK stage, avoiding characteristic decompositions while preserving accuracy and conservation. The method is validated through extensive Minkowski, axisymmetric cylindrical, and Kerr–Schild tests, including strong shocks, ultra-relativistic jets, and black-hole accretion, demonstrating robust stability, high-order accuracy, and effective constraint preservation in challenging GRHD scenarios.

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.

Figures (18)

• Figure 1: Example \ref{['1D Rie exam9']}: Numerical solutions at $t = 2$ obtained using the third-order PCP DG method with OE procedure (left) and without OE procedure (right) on a mesh of 200 uniform cells. Top: $\rho$; bottom: $p$.
• Figure 2: Example \ref{['1D Rie exam8']}: Close-up in $[0.5, 0.53]$ of the numerical solutions at $t = 0.43$, obtained using the third-order PCP DG method with OE procedure (left) and without OE procedure (right) on a mesh of 4000 uniform cells. Top: $\rho$; bottom: $p$.
• Figure 3: Example \ref{['1D Rie exam6']}: Numerical results for $\rho,\,v,\,p$ obtained using the $\mathbb{P}^m$-based PCP-OEDG method with 400 uniform cells at $t = 0.4$.
• Figure 4: Example \ref{['1D Rie exam7']}: Numerical results for $\rho,\,v,\,p$ obtained using the $\mathbb{P}^m$-based PCP-OEDG method with 800 uniform cells at $t = 0.45$.
• Figure 5: Example \ref{['1D Rie exam3']}: Numerical results for $\rho$ obtained using the $\mathbb{P}^m$-based OEDG method with 200 uniform cells at $t = 0.35$.

Theorems & Definitions (32)

• Lemma 2.1: Convexity W2017
• Remark 1
• Lemma 3.1: GQL representation
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Remark 2
• Lemma 3.4