An Algorithm for Computing Coefficients of Words in Expressions Involving Exponentials and its Application to the Construction of Exponential Integrators

TL;DR

This paper introduces a new algorithm for calculating word coefficients in exponential integrator expansions, enabling efficient construction of high-order methods with minimal exponentials.

Contribution

It presents a novel algorithm for computing coefficients of words in exponential integrator expansions, facilitating the design of high-order integrators with fewer exponentials.

Findings

Computed all 8th order self-adjoint Magnus-type integrators with minimal exponentials

Developed an efficient Maple implementation for order condition generation

Analyzed the structure of local error in exponential integrators

Abstract

This paper discusses an efficient implementation of the generation of order conditions for the construction of exponential integrators like exponential splitting and Magnus-type methods in the computer algebra system Maple. At the core of this implementation is a new algorithm for the computation of coefficients of words in the formal expansion of the local error of the integrator. The underlying theoretical background including an analysis of the structure of the local error is briefly reviewed. As an application the coefficients of all 8th order self-adjoint commutator-free Magnus-type integrators involving the minimum number of 8 exponentials are computed.