Alsallami, SAM, Niesen, J orcid.org/0000-0002-6693-3810 and Nijhoff, FW (2018) Closed-form modified Hamiltonians for integrable numerical integration schemes. Nonlinearity, 31 (11). pp. 5110-5146. ISSN 0951-7715
Abstract
Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually diverges. In contrast, this paper constructs and analyzes explicit examples of nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. We present cases of one- and twodegrees symplectic mappings arising as reductions of nonlinear integrable lattice equations, for which the modified Hamiltonians can be computed in closed form. These modified Hamiltonians are also given as power series in the time step by Yoshida’s method based on the Baker–Campbell–Hausdorff series. Another example displays an implicit dependence on the time step which could be of relevance to certain implicit schemes in numerical analysis. In light of these examples, the potential importance of integrable mappings to the field of geometric numerical integration is discussed.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Keywords: | modified Hamiltonian; symplectic integrators; integrable system; lattice KdV equation; mapping reduction; canonical transformation; separated variables |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 15 Aug 2018 13:49 |
Last Modified: | 10 Oct 2019 00:38 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1361-6544/aad9ac |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:134550 |