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

Xinyuan Wu () and Bin Wang
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Xinyuan Wu: Nanjing University, Department of Mathematics
Bin Wang: Qufu Normal University, School of Mathematical Sciences

Chapter Chapter 11 in Recent Developments in Structure-Preserving Algorithms for Oscillatory Differential Equations, 2018, pp 269-316 from Springer

Abstract: 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.

Date: 2018
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-10-9004-2_11

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DOI: 10.1007/978-981-10-9004-2_11

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