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Symmetry-Preserving Numerical Schemes

Chapter
Part of the CRM Series in Mathematical Physics book series (CRM)

Abstract

In these lectures we review two procedures for constructing finite difference numerical schemes that preserve symmetries of differential equations. The first approach is based on Lie’s infinitesimal symmetry generators, while the second method uses the novel theory of equivariant moving frames. The advantages of both techniques are discussed and illustrated with the Schwarzian differential equation, the Korteweg–de Vries equation and Burgers’ equation. Numerical simulations are presented and innovative techniques for obtaining better invariant numerical schemes are introduced. New research directions and open problems are indicated at the end of these notes.

Notes

Acknowledgements

The research of the first author is supported in part by a Tier 2 NSERC Canada Research Chair grant. The authors would like to thank the organizers of the ASIDE summer school for inviting them to give a series of lectures on continuous symmetries of discrete equations. We also thank Peter J. Olver for his comments on our lecture notes.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada
  2. 2.Department of MathematicsState University of New York at New PaltzNew PaltzUSA

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