We present a new, monolithic first--order (both in time and space) BSSNOK formulation of the coupled Einstein--Euler equations. The entire system of hyperbolic PDEs is solved in a completely unified manner via one single numerical scheme applied to both the conservative sector of the matter part and to the first--order strictly non--conservative sector of the spacetime evolution. The coupling between matter and space-time is achieved via algebraic source terms. The numerical scheme used for the solution of the new monolithic first order formulation is a path-conservative central WENO (CWENO) finite difference scheme, with suitable insertions to account for the presence of the non--conservative terms. By solving several crucial tests of numerical general relativity, including a stable neutron star, Riemann problems in relativistic matter with shock waves and the stable long-time evolution of single and binary puncture black holes up and beyond the binary merger, we show that our new CWENO scheme, introduced two decades ago for the compressible Euler equations of gas dynamics, can be successfully applied also to numerical general relativity, solving all equations at the same time with one single numerical method. In the future the new monolithic approach proposed in this paper may become an attractive alternative to traditional methods that couple central finite difference schemes with Kreiss-Oliger dissipation for the space-time part with totally different TVD schemes for the matter evolution and which are currently the state of the art in the field.

翻译：本文提出了一种新的、单片一阶（时间与空间均为一阶）BSSNOK形式的耦合爱因斯坦-欧拉方程组。该双曲型偏微分方程系统通过单一数值格式以完全统一的方式求解，该格式同时应用于物质部分的守恒部分和时空演化的一阶严格非守恒部分。物质与时空之间的耦合通过代数源项实现。用于求解该新型单片一阶形式的数值方案是一种路径守恒中心加权本质无振荡（CWENO）有限差分格式，其中针对非守恒项的存在进行了适当修正。通过求解数值广义相对论中的若干关键测试——包括稳定中子星、含激波的相对论性物质黎曼问题、以及单个及双穿刺黑洞在合并前后稳定长时间演化——我们证明，二十年前为气体动力学可压缩欧拉方程引入的CWENO格式，同样能成功应用于数值广义相对论，并使用单一数值方法同时求解所有方程。本文提出的新型单片方法未来可能成为传统方法的有力替代方案；传统方法通常对时空部分采用中心有限差分格式配合Kreiss-Oliger耗散，而对物质演化采用完全不同的总变差递减格式，这类方法目前仍是该领域的主流。