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Geometric Derivation and Analysis of Multi-Symplectic Numerical Schemes for Differential Equations

In: Computational Mathematics and Variational Analysis

Author

Listed:
  • Odysseas Kosmas

    (University of Manchester)

  • Dimitrios Papadopoulos

    (Delta Pi Systems Ltd.)

  • Dimitrios Vlachos

    (University of Peloponnese)

Abstract

In the current work we present a class of numerical techniques for the solution of multi-symplectic PDEs arising at various physical problems. We first consider the advantages of discrete variational principles and how to use them in order to create multi-symplectic integrators. We then consider the nonstandard finite difference framework from which these integrators derive. The latter is now expressed at the appropriate discrete jet bundle, using triangle and square discretization. The preservation of the discrete multi-symplectic structure by the numerical schemes is shown for several one- and two-dimensional test cases, like the linear wave equation and the nonlinear Klein–Gordon equation.

Suggested Citation

  • Odysseas Kosmas & Dimitrios Papadopoulos & Dimitrios Vlachos, 2020. "Geometric Derivation and Analysis of Multi-Symplectic Numerical Schemes for Differential Equations," Springer Optimization and Its Applications, in: Nicholas J. Daras & Themistocles M. Rassias (ed.), Computational Mathematics and Variational Analysis, pages 207-226, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-44625-3_12
    DOI: 10.1007/978-3-030-44625-3_12
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    Cited by:

    1. Odysseas Kosmas & Pieter Boom & Andrey P. Jivkov, 2021. "On the Geometric Description of Nonlinear Elasticity via an Energy Approach Using Barycentric Coordinates," Mathematics, MDPI, vol. 9(14), pages 1-16, July.

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