Symmetric spaces and Lie triple systems in numerical analysis of differential equations

TL;DR

This paper explores how symmetric spaces and Lie triple systems from differential geometry can unify and enhance numerical methods for differential equations, including new techniques for symmetry preservation in stiff problems.

Contribution

It introduces a mathematical framework linking geometric concepts to numerical algorithms and presents a novel Yoshida-like method to improve symmetry preservation in self-adjoint schemes.

Findings

Unified analysis of various numerical algorithms using geometric concepts

Development of a new technique to increase symmetry preservation order

Method particularly effective for stiff problems with positive time-steps

Abstract

A remarkable number of different numerical algorithms can be understood and analyzed using the concepts of symmetric spaces and Lie triple systems, which are well known in differential geometry from the study of spaces of constant curvature and their tangents. This theory can be used to unify a range of different topics, such as polar-type matrix decompositions, splitting methods for computation of the matrix exponential, composition of selfadjoint numerical integrators and dynamical systems with symmetries and reversing symmetries. The thread of this paper is the following: involutive automorphisms on groups induce a factorization at a group level, and a splitting at the algebra level. In this paper we will give an introduction to the mathematical theory behind these constructions, and review recent results. Furthermore, we present a new Yoshida-like technique, for self-adjoint numerical schemes, that allows to increase the order of preservation of symmetries by two units. Since all the time-steps are positive, the technique is particularly suited to stiff problems, where a negative time-step can cause instabilities.