On the iterated Crank-Nicolson for hyperbolic and parabolic equations in numerical relativity

TL;DR

This paper investigates the stability and properties of the iterated Crank-Nicolson scheme in numerical relativity, extending analysis to cases with unequal weights and parabolic equations, and proposes a variant with improved amplification and dispersion characteristics.

Contribution

It extends stability analysis of the iterated Crank-Nicolson scheme to new weighted averaging methods and parabolic equations, proposing a variant with enhanced numerical properties.

Findings

Unequal weights affect stability properties.

The proposed variant improves amplification factors.

The scheme reduces numerical dispersion.

Abstract

The iterated Crank-Nicolson is a predictor-corrector algorithm commonly used in numerical relativity for the solution of both hyperbolic and parabolic partial differential equations. We here extend the recent work on the stability of this scheme for hyperbolic equations by investigating the properties when the average between the predicted and corrected values is made with unequal weights and when the scheme is applied to a parabolic equation. We also propose a variant of the scheme in which the coefficients in the averages are swapped between two corrections leading to systematically larger amplification factors and to a smaller numerical dispersion.