Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System

TL;DR

This paper introduces a high-order, constraint-preserving discontinuous Galerkin method for simulating the spherically symmetric Einstein-Euler system, effectively handling physical constraints and geometric singularities with proven stability and accuracy.

Contribution

It develops a novel high-order CPcOEDG scheme that preserves physical and geometric constraints directly in conservative variables and metric potentials, improving robustness in relativistic fluid simulations.

Findings

Successfully simulates black hole accretion with stability.

Accurately captures relativistic shock waves.

Maintains physical realizability without primitive-variable checks.

Abstract

Numerical simulation of the spherically symmetric Einstein--Euler (EE) system faces severe challenges due to the stringent physical admissibility constraints of relativistic fluids and the geometric singularities inherent in metric evolution. This paper proposes a high-order Constraint-Preserving (CP) compact Oscillation-Eliminating Discontinuous Galerkin (cOEDG) method specifically tailored to address these difficulties. The method integrates a scale-invariant oscillation-eliminating mechanism [M. Peng, Z. Sun, K. Wu, Math. Comp., 94: 1147--1198, 2025] into a compact Runge--Kutta DG framework. By characterizing the convex invariant region of the hydrodynamic subsystem with general barotropic equations of state, we prove that the proposed scheme preserves physical realizability (specifically, positive density and subluminal velocity) directly in terms of conservative variables, thereby eliminating the need for complex primitive-variable checks. To ensure the geometric validity of the spacetime, we introduce a bijective transformation of the metric potentials. Rather than evolving the constrained metric components directly, the scheme advances unconstrained auxiliary variables whose inverse mapping automatically enforces strict positivity and asymptotic bounds without any limiters. Combined with a compatible high-order boundary treatment, the resulting CPcOEDG method exhibits robust stability and design-order accuracy in capturing strong gravity-fluid interactions, as demonstrated by simulations of black hole accretion and relativistic shock waves.