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Optimized high-order splitting methods for some classes of parabolic equations


Authors: S. Blanes, F. Casas, P. Chartier and A. Murua
Journal: Math. Comp. 82 (2013), 1559-1576
MSC (2010): Primary 65L05, 65P10, 37M15
DOI: https://doi.org/10.1090/S0025-5718-2012-02657-3
Published electronically: December 10, 2012
MathSciNet review: 3042575
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Abstract: We are concerned with the numerical solution obtained by splitting methods of certain parabolic partial differential equations. Splitting schemes of order higher than two with real coefficients necessarily involve negative coefficients. It has been demonstrated that this second-order barrier can be overcome by using splitting methods with complex-valued coefficients (with positive real parts). In this way, methods of orders $ 3$ to $ 14$ by using the Suzuki-Yoshida triple (and quadruple) jump composition procedure have been explicitly built. Here we reconsider this technique and show that it is inherently bounded to order $ 14$ and clearly sub-optimal with respect to error constants. As an alternative, we solve directly the algebraic equations arising from the order conditions and construct methods of orders $ 6$ and $ 8$ that are the most accurate ones available at present time, even when low accuracies are desired. We also show that, in the general case, 14 is not an order barrier for splitting methods with complex coefficients with positive real part by building explicitly a method of order $ 16$ as a composition of methods of order 8.


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Additional Information

S. Blanes
Affiliation: Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022 Valencia, Spain
Email: serblaza@imm.upv.es

F. Casas
Affiliation: Departament de Matemàtiques and IMAC, Universitat Jaume I, 12071 Castellón, Spain
Email: Fernando.Casas@mat.uji.es

P. Chartier
Affiliation: INRIA Rennes and Ecole Normale Supérieure de Cachan, Antenne de Bretagne, Avenue Robert Schumann, 35170 Bruz, France
Email: Philippe.Chartier@inria.fr

A. Murua
Affiliation: EHU/UPV, Konputazio Zientziak eta A.A. saila, Informatika Fakultatea, 12071 Donostia/San Sebastián, Spain
Email: Ander.Murua@ehu.es

DOI: https://doi.org/10.1090/S0025-5718-2012-02657-3
Keywords: Composition methods, splitting methods, complex coefficients, parabolic evolution equations
Received by editor(s): February 1, 2011
Received by editor(s) in revised form: October 13, 2011, and December 2, 2011
Published electronically: December 10, 2012
Additional Notes: The work of the first, second and fourth authors was partially supported by Ministerio de Ciencia e Innovación (Spain) under the coordinated project MTM2010-18246-C03 (co-financed by FEDER Funds of the European Union). Financial support from the “Acción Integrada entre España y Francia” HF2008-0105 was also acknowledged
The fourth author was additionally funded by project EHU08/43 (Universidad del País Vasco/Euskal Herriko Unibertsitatea).
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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