A monolithic first--order BSSNOK formulation of the Einstein--Euler equations and its solution with path-conservative finite difference CWENO schemes

TL;DR

The paper presents a monolithic first-order BSSNOK formulation for the Einstein–Euler system and a high-order path-conservative CWENO finite-difference scheme to solve the coupled equations with a single numerical method. By introducing 30 auxiliary gradient variables and a curl-free constraint term, the Einstein block becomes strongly hyperbolic and non-conservative, while the matter block remains conservative, enabling a unified discretization. The method achieves up to seventh-order accuracy and is validated on a broad suite of numerical relativity tests, including linear waves, hydrodynamic Riemann problems, TOV stars, and long-term evolutions of single and binary black holes, demonstrating robustness and potential as an alternative to traditional coupled schemes. The results suggest that a unified FO-BSSNOK–Euler solver with CWENO path-conservative discretization can handle complex general-relativistic flows and spacetime dynamics with high accuracy and stability, potentially informing future NR codes and simulations.

Abstract

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.

Figures (13)

• Figure 1: Linearized gravitational wave test solved with different version of the CWENO scheme. Left panel: $\tilde{A}_{zz}$ component of the extrinsic curvature at the final time, compared to the exact solution. Right panel: time evolution of the normalized Einstein constraints.
• Figure 2: Robust stability test case with a random initial perturbation of amplitude $10^{-7}/\rho^2$ in all quantities on a sequence of successively refined meshes on the unit square in 2D. The gamma--driver shift condition, $1+\log$ slicing and CWENO5 scheme have been used. Top left: $10\times10$ elements. Top right: $20\times20$ elements. Bottom left: $40\times40$ elements. Bottom right: $80\times80$ elements.
• Figure 3: Solution of the gauge wave test at $t=1000$ with $A=0.01$ using the CWENO7 scheme. Left panel: Evolution of the Einstein constraints. Right panel: profile of the metric term $\tilde{\gamma}_{xx}$ at the final time compared to the initial condition.
• Figure 4: Solution of Riemann Problem 1 at time $t=0.4$.
• Figure 5: Solution of Riemann Problem 2 at time $t=0.4$.