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爱因斯坦-欧拉方程的数值求解，基于Z4形式的一阶双曲重构，发展了一种带有子单元有限体积限制器的平衡不连续伽辽金有限元格式。我们分析并证明了由59个方程组成的一阶Z4系统对于一般度量具有强双曲性。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，实现了对任意但先验已知平衡态的平衡性。在数值黏性的设计上也特别考虑了平衡性质的保持。针对低密度物质处理（例如模拟被真空包围的中子星等大质量致密天体时），我们引入了一种新的滤波器用于守恒变量到原始变量的转换，当密度低于某一阈值时能防止超光速速度，这对于高度稀薄相对论天体物理流的数值研究也具有潜在的重要价值。得益于这些改进，所有数值相对论标准检验均成功复现，取得了三项成果：（i）能够对静止黑洞（包括具有极端自旋的克尔黑洞）进行稳定的长时间模拟，初始扰动后系统能以机器精度完美恢复平衡解；（ii）在纯真空条件（ρ=p=0）下对受扰动的（标准）TOV星进行演化至t=1000，无需在星体周围引入任何人为大气层；（iii）成功求解了两个穿孔黑洞的对头碰撞问题，该问题此前被认为在Z4框架下不可处理。