Splitting and composition methods in the numerical integration of differential equations

TL;DR

This paper surveys splitting and composition methods for numerically solving ordinary differential equations, highlighting their structure-preserving properties, backward error interpretation, and application examples.

Contribution

It provides a comprehensive overview of splitting and composition methods, including order conditions, scheme families, and practical illustrations.

Findings

Splitting methods are explicit and preserve geometric properties.

Composition methods can achieve high order accuracy.

Numerical examples demonstrate effectiveness in applications.

Abstract

We provide a comprehensive survey of splitting and composition methods for the numerical integration of ordinary differential equations (ODEs). Splitting methods constitute an appropriate choice when the vector field associated with the ODE can be decomposed into several pieces and each of them is integrable. This class of integrators are explicit, simple to implement and preserve structural properties of the system. In consequence, they are specially useful in geometric numerical integration. In addition, the numerical solution obtained by splitting schemes can be seen as the exact solution to a perturbed system of ODEs possessing the same geometric properties as the original system. This backward error interpretation has direct implications for the qualitative behavior of the numerical solution as well as for the error propagation along time. Closely connected with splitting integrators are composition methods. We analyze the order conditions required by a method to achieve a given order and summarize the different families of schemes one can find in the literature. Finally, we illustrate the main features of splitting and composition methods on several numerical examples arising from applications.