An Algorithm for Computing Coefficients of Words in Expressions Involving Exponentials and Its Application to the Construction of 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.Keywords: Splitting methods • Magnus-type integrators • Local error • Order conditions • Computer algebra. 4 For our considerations, it is sufficient to discuss linear problems. The algebraic structure underlying method construction is the same for nonlinear problems due to the calculus of Lie derivatives [8, Section III.5.1]. 5 We have used Maple 18, Maple is a trademark of Waterloo Maple Inc.

User manual for bch, a program for the fast computation of the Baker-Campbell-Hausdorff and similar series

“…In a preliminary step, the program computes the coefficients of all Lyndon words up to degree N in the power series of the given expression X of the form (3) using an adaptation of the algorithm from [9]. In the special case X = log(e A e A ) it alternatively uses the algorithm from the appendix of [7] and exploits the symmetry that the coefficient of the word…”

Adaptive Time Propagation for Time-dependent Schrödinger equations

“…One step of a CFM scheme for ( 1 ) starting at is defined by 2 with the ansatz [ 1 , 22 ] where the coefficients , are determined from the order conditions (a system of polynomial equations in the coefficients) such that the method attains convergence order p , see for example [ 19 ] and references therein. Algorithms to efficiently generate the order conditions are described for instance in [ 26 ]. Since such a system of equations generally does not define a unique solution, numerical optimization techniques are employed, for example minimizing the leading local error term of the resulting integrator.…”