A Parallelizable Implicit Evolution Scheme for Regge Calculus

TL;DR

This paper introduces a parallelizable implicit evolution scheme for Regge calculus, enabling efficient numerical relativity simulations by solving local equations at each vertex independently, demonstrated on a cosmological model.

Contribution

It presents a novel parallelizable implicit evolution scheme for Regge calculus that avoids global elliptic equations, improving computational efficiency in numerical relativity.

Findings

Scheme allows parallel vertex updates in Regge calculus.

Applied successfully to a Friedmann cosmology model.

Potential to make Regge calculus a practical tool in numerical relativity.

Abstract

The role of Regge calculus as a tool for numerical relativity is discussed, and a parallelizable implicit evolution scheme described. Because of the structure of the Regge equations, it is possible to advance the vertices of a triangulated spacelike hypersurface in isolation, solving at each vertex a purely local system of implicit equations for the new edge-lengths involved. (In particular, equations of global ``elliptic-type'' do not arise.) Consequently, there exists a parallel evolution scheme which divides the vertices into families of non-adjacent elements and advances all the vertices of a family simultaneously. The relation between the structure of the equations of motion and the Bianchi identities is also considered. The method is illustrated by a preliminary application to a 600--cell Friedmann cosmology. The parallelizable evolution algorithm described in this paper should enable Regge calculus to be a viable discretization technique in numerical relativity.