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Why Geometric Numerical Integration?

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Book cover Discrete Mechanics, Geometric Integration and Lie–Butcher Series

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 267))

Abstract

Geometric numerical integration (GNI) is a relatively recent discipline, concerned with the computation of differential equations while retaining their geometric and structural features exactly. In this paper we review the rationale for GNI and review a broad range of its themes: from symplectic integration to Lie-group methods, conservation of volume and preservation of energy and first integrals. We expand further on four recent activities in GNI: highly oscillatory Hamiltonian systems, W. Kahan’s ‘unconventional’ method, applications of GNI to celestial mechanics and the solution of dispersive equations of quantum mechanics by symmetric Zassenhaus splittings. This brief survey concludes with three themes in which GNI joined forces with other disciplines to shed light on the mathematical universe: abstract algebraic approaches to numerical methods for differential equations, highly oscillatory quadrature and preservation of structure in linear algebra computations.

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Notes

  1. 1.

    Occasionally known in the PDE literature as alternate direction methods.

  2. 2.

    Or, for that matter, a PDE, except that formalities are somewhat more complicated.

  3. 3.

    For traditional concepts such as Butcher tableaux, Runge-Kutta methods and B-series, the reader is referred to [34].

  4. 4.

    Note that in general \(S(\varvec{x})\) need not satisfy the so-called Jacobi identity.

  5. 5.

    A very readable early review of integrators for solar system dynamics is [63], cf also [62].

  6. 6.

    Also called equivariant.

  7. 7.

    To connect this to the GNI narrative, such a pattern is displayed by matrices in the symplectic Lie algebra \(\mathfrak {sp}(2n)\).

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Acknowledgements

This work has been supported by the Australian Research Council. The authors are grateful to David McLaren for assistance during the preparation of this paper, as well as to Philipp Bader, Robert McLachlan and Marcus Webb, whose comments helped to improve this paper.

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Authors and Affiliations

  1. Department of Applied Mathematics and Theoretical Physics Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK

    Arieh Iserles

  2. La Trobe University, Melbourne, Australia

    G. R. W. Quispel

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Correspondence to Arieh Iserles .

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    Kurusch Ebrahimi-Fard

  2. Institute for the Mathematical Sciences (ICMAT), Madrid, Spain

    María Barbero Liñán

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Iserles, A., Quispel, G.R.W. (2018). Why Geometric Numerical Integration?. In: Ebrahimi-Fard, K., Barbero Liñán, M. (eds) Discrete Mechanics, Geometric Integration and Lie–Butcher Series. Springer Proceedings in Mathematics & Statistics, vol 267. Springer, Cham. https://doi.org/10.1007/978-3-030-01397-4_1

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