Why Geometric Numerical Integration?

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


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.


Symplectic methods Lie-group methods Splittings Exponential integrators Kahan’s method Variational integrators Preservation of integrals Preservation of volume 

MSC numbers

Primary 65L05 Secondary 65P10 65M99 17B45 57S50 



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. 1.Department of Applied Mathematics and Theoretical Physics Centre for Mathematical SciencesUniversity of CambridgeCambridgeUK
  2. 2.La Trobe UniversityMelbourneAustralia

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