A New Optimality Property of Strang's Splitting

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 Störmer/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 $\dot{q}=0$, $\dot{p}=-Aq+f(q)$.

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