Commutator-free Lie group methods with minimum storage requirements and reuse of exponentials

TL;DR

This paper introduces a new class of commutator-free Lie group methods that minimize storage and reuse exponentials, based on explicit Runge-Kutta schemes, with proven results for third order and conjectures for higher orders.

Contribution

It proposes a novel format for Lie group methods that reduces storage requirements and reuses exponentials, supported by proofs and numerical evidence.

Findings

Proven for a 3-stage third order method.

Numerical examples support the conjecture for higher orders.

The methods are applicable to differential equations on manifolds.

Abstract

A new format for commutator-free Lie group methods is proposed based on explicit classical Runge-Kutta schemes. In this format exponentials are reused at every stage and the storage is required only for two quantities: the right hand side of the differential equation evaluated at a given Runge-Kutta stage and the function value updated at the same stage. The next stage of the scheme is able to overwrite these values. The result is proven for a 3-stage third order method and a conjecture for higher order methods is formulated. Five numerical examples are provided in support of the conjecture. This new class of structure-preserving integrators has a wide variety of applications for numerically solving differential equations on manifolds.