Monolithic first-order BSSNOK formulation of the Einstein-Euler equations and its solution with path-conservative finite difference central WENO schemes

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 Partial Differential Equations 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 nonconservative sector of the spacetime evolution. The coupling between matter and spacetime is achieved via algebraic source terms. The numerical scheme used for the solution of the new monolithic first-order formulation is a path-conservative finite difference central Weighted Essentially Non-Oscillatory (WENO) scheme, with suitable insertions to account for the presence of the nonconservative 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 longtime evolution of single- and binary-puncture black holes up and beyond the binary merger, we show that our new central WENO 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 spacetime part with totally different Total Variation Diminishing (TVD) schemes for the matter evolution and which are currently the state of the art in the field.