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

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.

Figures (16)

• Figure 1: Linearized gravitational wave test case solved with a fourth order ADER-DG scheme up to $t=1000$. Left panel: $\tilde{A}_{22}$ component of the conformal extrinsic curvature at the final time $t=1000$, compared to the exact solution. Right panel: time evolution of the Einstein constraints.
• Figure 2: Robust stability test for the FO-BSSNOK formulation with gamma--driver shift condition and $1+\log$ slicing. A random initial perturbation of amplitude $10^{-8}/\varrho^2$ in all quantities has been applied on a sequence of successively refined meshes on the unit square in 2D. The simulation has been carried out with a fourth order ADER-DG scheme ($N=3$). Top left: $10\times10$ elements ($\varrho=1$). Top right: $20\times20$ elements ($\varrho=2$). Bottom left: $30\times30$ elements ($\varrho=3$). Bottom right: $40\times40$ elements ($\varrho=4$).
• Figure 3: Solution of the gauge wave test with $A=0.01$ using a sixth order ADER-DG scheme ($N=5$). Left panel: Temporal evolution of the Einstein constraints. One observes the usual exponential growth of the constraints that is typical for the gauge wave test when applied to BSSNOK formulations of the Einstein equations. Right panel: profiles of the lapse $\alpha$ at $t=800$ and at $t=1000$ compared to the exact solution.
• Figure 4: Solution of Riemann Problem 1 at time $t=0.4$.
• Figure 5: Solution of Riemann Problem 2 at time $t=0.4$.