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Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs

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  • Islas, A.L.
  • Schober, C.M.

Abstract

Several recently developed multisymplectic schemes for Hamiltonian PDEs have been shown to preserve associated local conservation laws and constraints very well in long time numerical simulations. Backward error analysis for PDEs, or the method of modified equations, is a useful technique for studying the qualitative behavior of a discretization and provides insight into the preservation properties of the scheme. In this paper we initiate a backward error analysis for PDE discretizations, in particular of multisymplectic box schemes for the nonlinear Schrödinger equation. We show that the associated modified differential equations are also multisymplectic and derive the modified conservation laws which are satisfied to higher order by the numerical solution. Higher order preservation of the modified local conservation laws is verified numerically.

Suggested Citation

  • Islas, A.L. & Schober, C.M., 2005. "Backward error analysis for multisymplectic discretizations of Hamiltonian PDEs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 69(3), pages 290-303.
    • Handle: RePEc:eee:matcom:v:69:y:2005:i:3:p:290-303
    DOI: 10.1016/j.matcom.2005.01.006
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    Cited by:

    1. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
    2. Aydın, Ayhan, 2009. "Multisymplectic integration of N-coupled nonlinear Schrödinger equation with destabilized periodic wave solutions," Chaos, Solitons & Fractals, Elsevier, vol. 41(2), pages 735-751.
    3. Barletti, L. & Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2018. "Energy-conserving methods for the nonlinear Schrödinger equation," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 3-18.
    4. Frasca-Caccia, Gianluca & Hydon, Peter E., 2021. "Numerical preservation of multiple local conservation laws," Applied Mathematics and Computation, Elsevier, vol. 403(C).

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