Multi-block simulations in general relativity: high order discretizations, numerical stability, and applications

TL;DR

This paper introduces a high-order, stable finite difference method for multi-block simulations in general relativity, enabling accurate and stable modeling of complex spacetime geometries with multiple grids.

Contribution

It develops a high-order, stable finite difference scheme using summation by parts and penalty techniques for multi-grid general relativity simulations, ensuring numerical stability and accuracy.

Findings

Achieved stable high-order discretizations for multi-block grids.

Successfully applied methods to 1D, 2D, and 3D black hole models.

Demonstrated stability and accuracy in complex topologies.

Abstract

The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve the problem of interest under this grid-structure one must ensure the suitable transfer of information among the different grids involved. In this work we discuss a technique that allows one to construct finite difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The technique relies on the use of difference operators satisfying summation by parts and {\it penalty techniques} to transfer information between the grids. This allows the derivation of semidiscrete energy estimates for problems admitting such estimates at the continuum. We analyze several aspects of this technique when used in conjuction with high order schemes and illustrate its use in one, two and three dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision.