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Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

Author

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  • Cuicui Liao
  • Xiaohua Ding

Abstract

We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein‐Gordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.

Suggested Citation

  • Cuicui Liao & Xiaohua Ding, 2012. "Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    • Handle: RePEc:wly:jnljam:v:2012:y:2012:i:1:n:705179
    DOI: 10.1155/2012/705179
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    References listed on IDEAS

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    1. Khaled I. Nawafleh, 2011. "Canonical Quantization of Higher‐Order Lagrangians," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
    2. A. M. A. El-Sayed & A. Elsaid & D. Hammad, 2012. "A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-13, February.
    3. A. M. A. El-Sayed & A. Elsaid & D. Hammad, 2012. "A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein‐Gordon Equation of Arbitrary (Fractional) Orders," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    4. Xiao-Chuan Li & Wei-An Yao, 2011. "Symplectic Analytical Solutions for the Magnetoelectroelastic Solids Plane Problem in Rectangular Domain," Journal of Applied Mathematics, John Wiley & Sons, vol. 2011(1).
    5. Khaled I. Nawafleh, 2011. "Canonical Quantization of Higher-Order Lagrangians," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-11, October.
    6. Xiao-Chuan Li & Wei-An Yao, 2011. "Symplectic Analytical Solutions for the Magnetoelectroelastic Solids Plane Problem in Rectangular Domain," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-15, March.
    7. Rafael J. Villanueva & Abraham J. Arenas & Gilberto González-Parra, 2008. "A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics," Journal of Applied Mathematics, John Wiley & Sons, vol. 2008(1).
    8. Rafael J. Villanueva & Abraham J. Arenas & Gilberto González-Parra, 2008. "A Nonstandard Dynamically Consistent Numerical Scheme Applied to Obesity Dynamics," Journal of Applied Mathematics, Hindawi, vol. 2008, pages 1-14, December.
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    Cited by:

    1. Canan Koroglu & Ayhan Aydin, 2017. "An Unconventional Finite Difference Scheme for Modified Korteweg‐de Vries Equation," Advances in Mathematical Physics, John Wiley & Sons, vol. 2017(1).
    2. Mevlüde Yakıt Ongun & İlkem Turhan, 2013. "A Numerical Comparison for a Discrete HIV Infection of CD4+ T‐Cell Model Derived from Nonstandard Numerical Scheme," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).

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