Splitting Methods for differential equations

TL;DR

This survey reviews splitting methods for differential equations, discussing their order conditions, properties, and applications across various fields, including PDEs, celestial mechanics, and statistics.

Contribution

It provides a comprehensive analysis of splitting and composition methods, detailing their order conditions, qualitative properties, and practical applications in diverse scientific areas.

Findings

Detailed analysis of order conditions for splitting methods

Insights into geometric properties and oscillatory problems

Compilation and testing of methods across different orders

Abstract

This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyze in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.