Binary black hole simulation with an adaptive finite element method II: Application of local discontinuous Galerkin method to Einstein equations

TL;DR

This paper develops a finite element algorithm with local discontinuous Galerkin method for Einstein equations, demonstrating higher accuracy and shock capturing in spherical symmetric spacetime simulations compared to finite difference methods.

Contribution

It introduces a systematic finite element framework for Einstein equations, focusing on adaptive methods and local discontinuous Galerkin approach for binary compact objects.

Findings

Finite element method captures shock formation better than finite difference methods.

Finite element method shows higher accuracy in spherical symmetric spacetime simulations.

The approach is validated through numerical tests with simplified one-dimensional models.

Abstract

Finite difference method and pseudo-spectral method 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 much less 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 aim 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 the part reason for that the accuracy of finite element method is higher than finite difference method. At the same time kinds of formulation parameters setting are also discussed.