Setup of Order Conditions for Splitting Methods

Abstract

This article is based on [1] and [2], where an approach based on Taylor expansion and the structure of its leading term as an element of a free Lie algebra was described for the setup of a system of order conditions for operator splitting methods. Along with a brief review of these materials and some theoretical background, we discuss the implementation of the ideas from these papers in computer algebra, in particular using 3 Maple 18. A parallel version of such a code is described.

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

“…5 This can be realized in a very efficient way, utilizing a new algorithm derived in [9] for the computation of coefficients of words (i.e., finite products of non-commutative symbols) in expressions like (3) involving exponentials, whose concise Maple implementation is reproduced in its entirety in Section 2. 6 Our approach is thus a relevant contribution compared to previous work on the generation of order conditions, see, e.g., [8,Section III.5] or [3].…”

Numerical Simulation of Bloch Equations for Dynamic Magnetic Resonance Imaging

“…The detailed numerical analysis related to the advection-reaction equation is not presented further. For an in-depth discussion on the evolution equation with bounded operators, the reader may consult the review paper [92] by McLachlan et al and the book [38] by Farago et al There are a few recent studies on splitting techniques [60,59,6] which focus on the unbounded operators as well.…”