Operator Splitting Methods for Numerical Solutions of Ordinary Differential Equations

TL;DR

This paper develops operator-splitting methods for approximating Koopman generators of nonlinear dynamical systems, providing explicit error bounds and validating the approach through numerical experiments on classical systems.

Contribution

It introduces a framework for operator-splitting schemes applied to Koopman generators, with explicit error bounds and a bi-continuous Chernoff extension ensuring well-posedness.

Findings

Validated methods on Lotka-Volterra, Van der Pol, and Lorenz systems.

Demonstrated efficiency through work-precision comparisons.

Algorithms are simple, based on coordinate freezing and 1D solves.

Abstract

We study operator-splitting schemes for approximating Koopman generators of linear semigroups induced by nonlinear flows, a framework originating with Dorroh and Neuberger. Building on ideas of Lie, Kowalewski, and Gr\"{o}bner, we analyze the Koopman semigroup generated by the Lie-Koopman operator and exploit decompositions of this operator into finitely many components to construct Lie-Trotter, Strang, and higher-order compositions with explicit error bounds. A bi-continuous Chernoff extension guarantees well-posedness and contraction of the splitting operators. Numerical experiments on Lotka-Volterra, Van der Pol, and Lorenz systems validate the theory and demonstrate efficiency via work-precision comparisons. The algorithms remain conceptually simple, relying on coordinate freezing combined with one-dimensional solves, which reflects the classical separation-of-variables principle.