Application of local discontinuous Galerkin method to Einstein equations

Abstract

Finite difference and pseudo-spectral methods have been widely used in the numerical relativity to solve the Einstein equations. As the third major category method to solve partial differential equations, finite element method is less frequently used in numerical relativity. In this paper, we design a finite element algorithm to solve the evolution part of the Einstein equations. This paper is the second one of a systematic investigation of applying adaptive finite element method to the Einstein equations, especially aiming for binary compact objects simulations. The first paper of this series has been contributed to the constrained part of the Einstein equations for initial data. Since applying finite element method to the Einstein equations is a big project, we mainly propose the theoretical framework of a finite element algorithm together with local discontinuous Galerkin method for the Einstein equations in the current work. In addition, we have tested our algorithm based on the spherical symmetric spacetime evolution. In order to simplify our numerical tests, we have reduced the problem to a one-dimensional space problem by taking the advantage of the spherical symmetry. Our reduced equation system is a new formalism for spherical symmetric spacetime simulation. Based on our test results, we find that our finite element method can capture the shock formation which is introduced by numerical error. In contrast, such shock is smoothed out by numerical dissipation within the finite difference method. We suspect this is partly the reason that the accuracy of finite element method is higher than the finite difference method. At the same time, different kinds of formulation parameters setting are also discussed.

GRChombo: An adaptable numerical relativity code for fundamental physics

“…Other notable but non-public codes include BAM (Bruegmann et al, 2008;Marronetti et al, 2007), AMSS-NCKU (Galaviz et al, 2010), PAMR/AMRD and HAD (East et al, 2012;Neilsen et al, 2007). Codes such as SPeC (Pfeiffer et al, 2003) and bamps (Hilditch et al, 2016) implement the generalised harmonic formulation of the Einstein equations using a pseudospectral method, and discontinuous Galerkin methods are used in SpECTRE (https://spectre-code.org) (Deppe et al, 2021;Kidder & others, 2017) (see also (Cao et al, 2018)). NRPy (http: //astro.phys.wvu.edu/bhathome) (Ruchlin et al, 2018) is a code aimed for use on non-HPC systems, which generate C code from Python, and uses adapted coordinate systems to minimise computational costs.…”

The Gravitational-wave physics II: Progress

“…After the initial data are ready, we come to the evolution equations [735]. There are well developed finite element methods for Hamilton-Jacobi-like equations [736]…”

“…where the unknown variables u μ and the related A μ ν (u σ ) and S μ (u σ ) are given in ref. [735]. Eq.…”

Lessons for adaptive mesh refinement in numerical relativity

“…Alternatives are problem-adapted coordinate systems, e.g. NRPy+ [68] or discontinuous Galerkin methods as in SpECTRE [69,70] (see also [71,72]). A brief overview of the history of NR codes can be found in [73].…”