High-order discontinuous Galerkin schemes with subcell finite volume limiter and adaptive mesh refinement for a monolithic first-order BSSNOK formulation of the Einstein-Euler equations

TL;DR

This paper introduces a high-order discontinuous Galerkin scheme with a subcell finite volume limiter and adaptive mesh refinement for solving the coupled Einstein-Euler equations in numerical relativity, demonstrating stability and accuracy in complex simulations.

Contribution

It presents a novel high-order DG scheme with a subcell FV limiter and AMR for the monolithic Einstein-Euler system, enabling stable long-term black hole merger simulations.

Findings

Successful long-term evolution of black hole mergers

Good agreement with reference solutions

Robust handling of shocks and singularities

Abstract

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.