保持约束的高阶紧凑OEDG方法用于球对称的爱因斯坦-欧拉系统
Constraint-Preserving High-Order Compact OEDG Method for Spherically Symmetric Einstein-Euler System
AI总结：
本文提出了一种高阶紧凑OEDG方法，用于解决球对称爱因斯坦-欧拉系统中严格物理约束和几何奇点的问题，通过引入尺度不变振荡消除机制和双射变换，实现了高阶精度和稳定性。
摘要：
球对称的爱因斯坦-欧拉（EE）系统数值模拟面临严峻挑战，因为相对流体的严格物理可接受性约束和度量演化的几何奇点。本文提出了一种高阶约束保持（CP）紧凑振荡消除不连续伽辽金（cOEDG）方法，专门针对这些困难。该方法将一种尺度不变的振荡消除机制[刘鹏，孙泽，吴凯，Math. Comp., 94: 1147--1198, 2025]整合到紧凑的Runge-Kutta DG框架中。通过使用一般等熵方程状态来表征流体子系统的凸不变区域，我们证明所提出的方案直接在守恒变量中保持物理可实现性（即正密度和次光速速度），从而消除了复杂原始变量检查的需要。为了确保时空的几何有效性，我们引入了度量势的双射变换。而不是直接演进度量约束成分，该方案推进了无约束的辅助变量，其反向映射自动强制严格正性和渐近界，而无需任何限制器。结合兼容的高阶边界处理，所得到的CPcOEDG方法在捕捉强引力-流体相互作用方面表现出稳健的稳定性和设计阶精度，如黑洞吸积和相对论冲击波的模拟所证明的。
英文摘要：
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.
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