A spectral approach to numerical simulations of the ADM equations

TL;DR

This paper develops a spectral numerical method for simulating vacuum Einstein equations in ADM formalism, incorporating adaptive time refinement and filtering to enhance stability and accuracy, validated through standard tests and new initial data.

Contribution

Introduces a spectral approach with adaptive time refinement and filtering for stable, accurate ADM equation simulations in vacuum, including new initial data for testing.

Findings

Validated the spectral method with standard tests.

Achieved stable simulations with adaptive time refinement.

Provided new initial data for numerical code testing.

Abstract

We present a numerical study of the Einstein equations, according to the Arnowitt-Deser-Misner (ADM) formalism, in order to simulate the dynamics of gravitational fields. We took in consideration the original $3+1$ decomposition of the ADM equations, in vacuum conditions, in simplified geometries. The numerical code is based on spectral methods, making use of filtering (de-aliasing) techniques. The algorithm has been stabilized via an adaptive time-refinement, based on a procedure that checks self-consistently the regularity of the solutions. The accuracy of our numerical model has been validated through a series of standard tests. Finally, we present also a new kind of initial data that can be used for testing numerical codes.