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

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.

Incorporated referee comments during peer-review. Added several references pointed out to us. Added a figure and discussion comparing pointwise error for 4th- and 5th-order stencils for wave equation

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• AsterX: a new open-source GPU-accelerated GRMHD code for dynamical spacetimes
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• Extremely High-Order Convergence in Simulations of Relativistic Stars