Local error structures and order conditions in terms of Lie elements for exponential splitting schemes

Auzinger, Winfried; Herfort, Wolfgang

doi:10.7494/opmath.2014.34.2.243

Cited by 20 publications

(57 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As described in [9], this can be used to set up a recursive algorithm for the generation of order conditions. Furthermore, for a given scheme of order p, the coefficients in the linear combination (2.10) can then be computed from the conditions for order p + 1.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…According to [9], for a general splitting method of order p the leading term L 0 (t) has a special structure, namely L 0 (t) = linear combination of p -th iterated commutators of A, B, C , (2.10)…”

Section: Notation and Preliminariesmentioning

confidence: 99%

“…Higher-order schemes can be constructed via composition. The construction of optimized higher-order schemes requires the solution of a nonlinear system of order conditions as for instance described in [9].…”

Section: Introductionmentioning

confidence: 99%

“…To this end the defect-based approach developed in [9,[11][12][13] is extended to the case of three operators. This idea is rather universally applicable to one-step schemes, and it is of particular interest in the present context because other ways of estimating the error, for example using pairs of embedded formulas as proposed in [10], are difficult to realize and tend to be inefficient whenever three sub-operators are involved and higher-order schemes, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…To create a framework for this nontrivial extension of our prior approach [9,[11][12][13], we consider an abstract linear evolution equation where the operator on the righthand side is split into three parts,…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Defect-based local error estimators for high-order splitting methods involving three linear operators

2014

Self Cite

View full text Add to dashboard Cite

Prior work on high-order exponential operator splitting methods is extended to evolution equations defined by three linear operators. A posteriori local error estimators are constructed via a suitable integral representation of the local error involving the defect associated with the splitting solution and quadrature approximation via Hermite interpolation. In order to prove asymptotical correctness, a multiple integral representation involving iterated defects is deduced by repeated application of the variation-of-constant formula. The error analysis within the framework of abstract evolution equations provides the basis for concrete applications. Numerical examples for initial-boundary value problems of Schrödinger and of parabolic type confirm the asymptotical correctness of the proposed a posteriori error estimators.Keywords Linear evolution equations · Time integration methods · High-order exponential operator splitting methods · Local error · A posteriori local error estimators Mathematics Subject Classification (2010) 65J10 · 65L05 · 65M12 · 65M15

show abstract

Section: Notation and Preliminariesmentioning

confidence: 99%

Section: Notation and Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Defect-based local error estimators for high-order splitting methods involving three linear operators

2014

Self Cite

View full text Add to dashboard Cite

show abstract

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

Hofstätter¹,

Auzinger

Koch

2019

Computer Algebra in Scientific Computing

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Setup of Order Conditions for Splitting Methods

Auzinger

Herfort

Hofstätter

et al. 2016

Computer Algebra in Scientific Computing

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Local error structures and order conditions in terms of Lie elements for exponential splitting schemes

Cited by 20 publications

References 11 publications

Defect-based local error estimators for high-order splitting methods involving three linear operators

Defect-based local error estimators for high-order splitting methods involving three linear operators

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

Setup of Order Conditions for Splitting Methods

Contact Info

Product

Resources

About