An almost symmetric Strang splitting scheme for the construction of high order composition methods

TL;DR

This paper introduces an almost symmetric Strang splitting scheme that enables high order composition methods for nonlinear ODEs where exact flows are infeasible, demonstrating improved efficiency and conservation properties.

Contribution

It develops an iterated Strang splitting scheme that is symmetric up to a certain order, facilitating high order composition methods for complex nonlinear problems.

Findings

Achieves high order accuracy up to order six.

Demonstrates superior efficiency over standard methods.

Shows better conservation properties in numerical experiments.

Abstract

In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure can not be computed exactly. Instead, we insert a well-chosen state $y_{\star}$ into the corresponding nonlinearity $b(y)y$, which results in a linear term $b(y_{\star})y$ whose exact flow can be determined efficiently. Therefore, in the spirit of splitting methods, it is still possible for the numerical simulation to satisfy certain properties of the exact flow. However, Strang splitting is no longer symmetric (even though it is still a second order method) and thus high order composition methods are not easily attainable. We will show that an iterated Strang splitting scheme can be constructed which yields a method that is symmetric up to a given order. This method can then be used to attain high order composition schemes. We will illustrate our theoretical results, up to order six, by conducting numerical experiments for a charged particle in an inhomogeneous electric field, a post-Newtonian computation in celestial mechanics, and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to Strang splitting. For the first example we also perform a comparison with the standard fourth order Runge--Kutta methods and find significant gains in efficiency as well better conservation properties.