An Operator-Based Local Discontinuous Galerkin Method Compatible With the BSSN Formulation of the Einstein Equations

TL;DR

This paper introduces an operator-based local discontinuous Galerkin method compatible with the BSSN formulation of Einstein equations, enabling stable, efficient, and parallelizable solutions for complex 3+1 dimensional relativistic problems.

Contribution

It generalizes local DG methods to solve second-order hyperbolic equations by discretizing the derivative operator, bridging DGFE and finite difference approaches.

Findings

Enables solving full 3+1 dimensional BSSN equations with DGFE methods.

Provides a discretization approach that is both flux-conservative and operator-based.

Facilitates stable puncture-type evolutions of black hole systems.

Abstract

Discontinuous Galerkin Finite Element (DGFE) methods offer a mathematically beautiful, computationally efficient, and efficiently parallelizable way to solve hyperbolic partial differential equations. These properties make them highly desirable for numerical calculations in relativistic astrophysics and many other fields. The BSSN formulation of the Einstein equations has repeatedly demonstrated its robustness. The formulation is not only stable but allows for puncture-type evolutions of black hole systems. To-date no one has been able to solve the full (3+1)-dimensional BSSN equations using DGFE methods. This is partly because DGFE discretization often occurs at the level of the equations, not the derivative operator, and partly because DGFE methods are traditionally formulated for manifestly flux-conservative systems. By discretizing the derivative operator, we generalize a particular flavor of DGFE methods, Local DG methods, to solve arbitrary second-order hyperbolic equations. Because we discretize at the level of the derivative operator, our method can be interpreted as either a DGFE method or as a finite differences stencil with non-constant coefficients.