In this paper we develop a new well-balanced discontinuous Galerkin (DG) finite element scheme with subcell finite volume (FV) limiter for the numerical solution of the Einstein--Euler equations of general relativity based on a first order hyperbolic reformulation of the Z4 formalism. The first order Z4 system, which is composed of 59 equations, is analyzed and proven to be strongly hyperbolic for a general metric. The well-balancing is achieved for arbitrary but a priori known equilibria by subtracting a discrete version of the equilibrium solution from the discretized time-dependent PDE system. Special care has also been taken in the design of the numerical viscosity so that the well-balancing property is achieved. As for the treatment of low density matter, e.g. when simulating massive compact objects like neutron stars surrounded by vacuum, we have introduced a new filter in the conversion from the conserved to the primitive variables, preventing superluminal velocities when the density drops below a certain threshold, and being potentially also very useful for the numerical investigation of highly rarefied relativistic astrophysical flows. Thanks to these improvements, all standard tests of numerical relativity are successfully reproduced, reaching three achievements: (i) we are able to obtain stable long term simulations of stationary black holes, including Kerr black holes with extreme spin, which after an initial perturbation return perfectly back to the equilibrium solution up to machine precision; (ii) a (standard) TOV star under perturbation is evolved in pure vacuum ($\rho=p=0$) up to $t=1000$ with no need to introduce any artificial atmosphere around the star; and, (iii) we solve the head on collision of two punctures black holes, that was previously considered un--tractable within the Z4 formalism.

翻译：本文针对广义相对论中的爱因斯坦-欧拉方程，基于Z4形式的一阶双曲重构，发展了一种具有子单元有限体积限制器的新型平衡保持间断伽辽金有限元格式。通过对包含59个方程的一阶Z4系统进行分析，证明了该系统在一般度规下具有强双曲性。平衡特性通过从离散化的含时偏微分方程系统中减去平衡解的离散版本实现，适用于任意但事先已知的平衡态。数值粘性的设计也经过特殊处理以确保平衡保持性质。针对低密度物质（例如模拟中子星等大质量致密天体被真空包围的场景）的处理，我们在守恒变量到原始变量的转换中引入了新型滤波器，避免密度低于特定阈值时出现超光速速度，该技术对高度稀薄相对论天体物理流的数值研究亦具有重要应用价值。得益于这些改进，所有数值相对论标准测试均成功复现，实现了三项突破：（i）能够对静止黑洞（包括极端自旋的克尔黑洞）进行稳定的长期模拟，初始扰动后系统能精确恢复到机器精度的平衡解；（ii）在纯真空条件（ρ=p=0）下对受扰动的TOV恒星进行演化模拟至t=1000，无需在星体周围引入任何人造大气层；（iii）成功求解了此前被认为在Z4框架下难以处理的双虫洞黑洞对头碰撞问题。