Regge Calculus as a Fourth Order Method in Numerical Relativity

TL;DR

This paper investigates the convergence properties of Regge calculus in numerical relativity, demonstrating that averaging over local equations improves convergence order from second to fourth power of lattice spacing.

Contribution

It analytically and numerically shows that averaging Regge equations enhances convergence order from second to fourth power in numerical relativity.

Findings

Individual Regge equations converge as second order

Averaged equations converge as third order analytically

Numerical results show fourth order convergence

Abstract

The convergence properties of numerical Regge calculus as an approximation to continuum vacuum General Relativity is studied, both analytically and numerically. The Regge equations are evaluated on continuum spacetimes by assigning squared geodesic distances in the continuum manifold to the squared edge lengths in the simplicial manifold. It is found analytically that, individually, the Regge equations converge to zero as the second power of the lattice spacing, but that an average over local Regge equations converges to zero as (at the very least) the third power of the lattice spacing. Numerical studies using analytic solutions to the Einstein equations show that these averages actually converge to zero as the fourth power of the lattice spacing.