Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System
Yuchen Huang, Manting Peng, Kailiang Wu
TL;DR
This work develops a high-order, constraint-preserving CPcOEDG method for the spherically symmetric Einstein–Euler system, addressing both relativistic fluid admissibility and geometric metric constraints. By proving a convex, conservative-variable representation of physical states and evolving metric auxiliary variables via bijections, the scheme enforces positivity and subluminality without primitive-variable checks or limiter-driven distortions. The method integrates a scale-invariant oscillation-eliminating filter into a compact RKDG framework and leverages Geometric Quasi-Linearization flux inequalities to guarantee stability and design-order accuracy under suitable CFL conditions. Numerical experiments across FRW, TOV, and black-hole accretion scenarios confirm high-order convergence, robust stability, and sharp shock-capturing without spurious oscillations, highlighting the approach’s potential for realistic relativistic astrophysical simulations.
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.