A New Optimality Property of Strang's Splitting

TL;DR

This paper demonstrates that Strang's splitting method possesses an optimal stability property for certain systems, extending known stability results of Verlet/Leapfrog algorithms to more general splitting schemes.

Contribution

It establishes that Strang splitting is optimally stable among splitting integrators for a class of systems, generalizing classical stability properties.

Findings

Strang splitting has a larger stability region than alternatives.

The stability property extends from linear to nonlinear split systems.

Verlet/Leapfrog stability is a special case of this general property.

Abstract

For systems of the form $\dot q = M^{-1} p$, $\dot p = -Aq+f(q)$, common in many applications, we analyze splitting integrators based on the (linear/nonlinear) split systems $\dot q = M^{-1} p$, $\dot p = -Aq$ and $\dot q = 0$, $\dot p = f(q)$. We show that the well-known Strang splitting is optimally stable in the sense that, when applied to a relevant model problem, it has a larger stability region than alternative integrators. This generalizes a well-known property of the common St\"{o}rmer/Verlet/leapfrog algorithm, which of course arises from Strang splitting based on the (kinetic/potential) split systems $\dot q = M^{-1} p$, $\dot p = 0$ and $\dot q = 0$, $\dot p = -Aq+f(q)$.