Finite Element, Discontinuous Galerkin, and Finite Difference Evolution Schemes in Spacetime

Abstract

Numerical schemes for Einstein's vacuum equation are developed. Einstein's equation in harmonic gauge is second order symmetric hyperbolic. It is discretized in four-dimensional spacetime by Finite Differences, Finite Elements, and Interior Penalty Discontinuous Galerkin methods, the latter related to Regge calculus. The schemes are split into space and time and new time-stepping schemes for wave equations are derived. The methods are evaluated for linear and non-linear test problems of the Apples-with-Apples collection.

final version

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Cited by in corpus (15)

• Continuum and Discrete Initial-Boundary-Value Problems and Einstein's Field Equations
• SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics
• Solving the relativistic magnetohydrodynamics equations with ADER discontinuous Galerkin methods, a posteriori subcell limiting and adaptive mesh refinement
• Discontinuous Galerkin methods for general-relativistic hydrodynamics: formulation and application to spherically symmetric spacetimes
• Numerical simulations of high Lundquist number relativistic magnetic reconnection
• Solving 3D relativistic hydrodynamical problems with WENO discontinuous Galerkin methods
• Formulation of discontinuous Galerkin methods for relativistic astrophysics
• Constraint Propagation of $C^2$-adjusted Formulation - Another Recipe for Robust ADM Evolution System
• General-relativistic neutron star evolutions with the discontinuous Galerkin method
• The Final Merger of Black-Hole Binaries
• An Operator-Based Local Discontinuous Galerkin Method Compatible With the BSSN Formulation of the Einstein Equations
• Simulating magnetized neutron stars with discontinuous Galerkin methods
• A hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity
• Constraint Propagation of $C^2$-adjusted Formulation II --- Another Recipe for Robust Baumgarte-Shapiro-Shibata-Nakamura Evolution System ---
• Fourth order indirect integration method for black hole perturbations: even modes