We propose a high order discontinuous Galerkin (DG) scheme with subcell finite volume (FV) limiter to solve a monolithic first--order hyperbolic BSSNOK formulation of the coupled Einstein--Euler equations. The numerical scheme runs with adaptive mesh refinement (AMR) in three space dimensions, is endowed with time-accurate local time stepping (LTS) and is able to deal with both conservative and non-conservative hyperbolic systems. The system of governing partial differential equations was shown to be strongly hyperbolic and is solved in a monolithic fashion with one numerical framework that can be simultaneously applied to both the conservative matter subsystem as well as the non-conservative subsystem for the spacetime. Since high order unlimited DG schemes are well-known to produce spurious oscillations in the presence of discontinuities and singularities, our subcell finite volume limiter is crucial for the robust discretization of shock waves arising in the matter as well as for the stable treatment of puncture black holes. We test the new method on a set of classical test problems of numerical general relativity, showing good agreement with available exact or numerical reference solutions. In particular, we perform the first long term evolution of the inspiralling merger of two puncture black holes with a high order ADER-DG scheme.

翻译：本文提出一种结合子单元有限体积限制器的高阶间断伽辽金格式，用于求解耦合爱因斯坦-欧拉方程组的单体一阶双曲型BSSNOK表述。该数值格式在三维空间中采用自适应网格细化技术运行，具备时间精确的局部时间步进特性，能够同时处理守恒型与非守恒型双曲系统。该控制偏微分方程组已被证明为强双曲型，并通过单一数值框架以单体方式进行求解，该框架可同时应用于守恒型物质子系统与非守恒型时空子系统。由于无限制的高阶间断伽辽金格式在间断和奇点附近会产生伪振荡，本文采用的子单元有限体积限制器对于物质中激波现象的鲁棒性离散化以及穿刺黑洞的稳定处理至关重要。我们在数值广义相对论的一系列经典测试问题上验证了新方法，结果显示与现有精确解或数值参考解高度吻合。特别地，我们首次采用高阶ADER-DG格式实现了两个穿刺黑洞旋近合并的长期演化模拟。