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Splitting methods for differential equations

Part of: Numerical analysis: Ordinary differential equations Numerical problems in dynamical systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  04 September 2024

Sergio Blanes Open the ORCID record for Sergio Blanes [Opens in a new window] ,
Fernando Casas  and
Ander Murua
Sergio Blanes
Affiliation:
Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, 46022-Valencia, Spain E-mail: serblaza@imm.upv.es
Fernando Casas
Affiliation:
Departament de Matemàtiques and IMAC, Universitat Jaume I, 12071-Castellón, Spain E-mail: Fernando.Casas@uji.es
Ander Murua
Affiliation:
Konputazio Zientziak eta A.A. Saila, Informatika Fakultatea, EHU/UPV, Donostia/San Sebastián, Spain E-mail: Ander.Murua@ehu.es
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Abstract

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This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class of integrators are composition methods, in which one or several low-order schemes are composed to construct higher-order numerical approximations to the exact solution. We analyse in detail the order conditions that have to be satisfied by these classes of methods to achieve a given order, and provide some insight about their qualitative properties in connection with geometric numerical integration and the treatment of highly oscillatory problems. Since splitting methods have received considerable attention in the realm of partial differential equations, we also cover this subject in the present survey, with special attention to parabolic equations and their problems. An exhaustive list of methods of different orders is collected and tested on simple examples. Finally, some applications of splitting methods in different areas, ranging from celestial mechanics to statistics, are also provided.

MSC classification

Primary: 65L05: Initial value problems 65L20: Stability and convergence of numerical methods 65P10: Hamiltonian systems including symplectic integrators
Secondary: 65M12: Stability and convergence of numerical methods 65M70: Spectral, collocation and related methods
Type
Research Article
Information
Acta Numerica , Volume 33 , July 2024 , pp. 1 - 161
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2024. Published by Cambridge University Press

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