On the Stability of the Iterated Crank-Nicholson Method in Numerical Relativity

TL;DR

This paper demonstrates that in numerical relativity, performing exactly two iterations of the iterated Crank-Nicholson method is optimal, as more iterations do not improve stability and can worsen results.

Contribution

It reveals that only two iterations of the iterated Crank-Nicholson method should be used for stability, challenging the assumption that more iterations are better.

Findings

Two iterations are optimal for stability.

More than two iterations can worsen results.

Infinite iterations converge to standard Crank-Nicholson, which is less stable.

Abstract

The iterated Crank-Nicholson method has become a popular algorithm in numerical relativity. We show that one should carry out exactly two iterations and no more. While the limit of an infinite number of iterations is the standard Crank-Nicholson method, it can in fact be worse to do more than two iterations, and it never helps. We explain how this paradoxical result arises.