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Backward error analysis for multi-symplectic integration methods

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Summary.

A useful method for understanding discretization error in the numerical solution of ODEs is to compare the system of ODEs with the modified equations obtained through backward error analysis, and using symplectic integration for Hamiltonian ODEs provides more incite into the modified equations. In this paper, the ideas of symplectic integration are extended to Hamiltonian PDEs, and this paves the way for the development of a local modified equation analysis solely as a useful diagnostic tool for the study of these types of discretizations. In particular, local conservation laws of energy and momentum are not preserved exactly when symplectic integrators are used to discretize, but the modified equations are used to derive modified conservation laws that are preserved to higher order along the numerical solution. These results are also applied to the nonlinear wave equation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Surrey, and Department of Mathematics, Imperial College, London

    Brian Moore

  2. Department of Mathematics, Imperial College, London

    Sebastian Reich

Authors
  1. Brian Moore
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  2. Sebastian Reich
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Correspondence to Sebastian Reich.

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Mathematics Subject Classification (1991): 65M06, 65P10, 37K05

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Moore, B., Reich, S. Backward error analysis for multi-symplectic integration methods. Numer. Math. 95, 625–652 (2003). https://doi.org/10.1007/s00211-003-0458-9

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  • DOI: https://doi.org/10.1007/s00211-003-0458-9

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