Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case

“…For a detailed analysis for the nonlinear case in a simpler setting (splitting into two operators and low order methods) we refer to [11]. Here we illustrate the algorithmic evaluation of D(t, u) for the case s = 3, with obvious generalization to general sstage schemes.…”

Section: The Nonlinear Casementioning

confidence: 99%

“…As this structure cannot be reduced to the case of two operators, an extended framework for the analysis needs to be created. We provide a complete analysis for linear problems; the general ideas related to the local error structure and a posteriori error estimation are the same in the nonlinear case, but technicalities abound, see for instance the discussion of splitting into two operators in [11]. Within our abstract setting we are not specific about the underlying function spaces.…”

Section: Introductionmentioning

confidence: 98%

“…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 2 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

Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators

Auzinger

Hofstätter

Koch

2016

Computer Algebra in Scientific Computing

Self Cite

View full text Add to dashboard Cite

We describe a package realized in the Julia programming language which performs symbolic manipulations applied to nonlinear evolution equations, their flows, and commutators of such objects. This tool was employed to perform contrived computations arising in the analysis of the local error of operator splitting methods. It enabled the proof of the convergence of the basic method and of the asymptotical correctness of a defect-based error estimator. The performance of our package is illustrated on several examples. Problem settingWe are interested in the solution to nonlinear evolution equationson a Banach space X, where A and B are general nonlinear, unbounded operators defined on a subset D ⊂ X, the solution is denoted by E H (t, u 0 ), and analogously for the two sub-flows associated with A and B. The structure of the vector fields often suggests to employ additive splitting methods to separately propagate the two subproblems defined by A and B,

show abstract

Defect-based local error estimators for splitting methods, with application to Schrödinger equations, Part III: The nonlinear case

Cited by 15 publications

References 14 publications

Representation of the local error for higher-order exponential splitting schemes involving two or three sub-operators

Representation of the local error for higher-order exponential splitting schemes involving two or three sub-operators

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

Symbolic Manipulation of Flows of Nonlinear Evolution Equations, with Application in the Analysis of Split-Step Time Integrators

Contact Info

Product

Resources

About