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Arbitrarily High-Order Time-Stepping Schemes for Nonlinear Klein–Gordon Equations

In: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations

Author

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  • Xinyuan Wu

    (Nanjing University, Department of Mathematics
    Qufu Normal University, School of Mathematical Sciences)

  • Bin Wang

    (Qufu Normal University, School of Mathematical Sciences)

Abstract

This chapter presentsArbitrarily high–order time–stepping methods arbitrarily high-order time-stepping schemes for solving high-dimensional nonlinear Klein–Gordon equations with different boundary conditions. We first formulate an abstract ordinary differential equation (ODE) on a suitable infinite–dimensional function space based on the operator spectrum theory. We then introduce an operator-variation-of-constants formula for the nonlinear abstract ODE. The nonlinear stability and convergence are rigorously analysed once the spatial differential operator is approximated by an appropriate positive semi-definite matrix. With regard to the two dimensional Dirichlet or Neumann boundary problems, the time-stepping schemes coupled with discrete Fast Sine/Cosine Transformation can be applied to simulate the two-dimensional nonlinear Klein–Gordon equations effectively. The numerical results demonstrate the advantage of the schemes in comparison with the existing numerical methods for solving nonlinear Klein–Gordon equations in the literature.

Suggested Citation

  • Xinyuan Wu & Bin Wang, 2018. "Arbitrarily High-Order Time-Stepping Schemes for Nonlinear Klein–Gordon Equations," Springer Books, in: Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations, chapter 0, pages 269-316, Springer.
    • Handle: RePEc:spr:sprchp:978-981-10-9004-2_11
    DOI: 10.1007/978-981-10-9004-2_11
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