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High-Order Symmetric Hermite–Birkhoff Time Integrators for Semilinear KG Equations

In: Geometric Integrators for Differential Equations with Highly Oscillatory Solutions

Author

Listed:
  • Xinyuan Wu

    (Nanjing University, Department of Mathematics)

  • Bin Wang

    (Xi’an Jiaotong University, School of Mathematics and Statistics)

Abstract

The computation of the Klein–Gordon equation featuring a nonlinear potential function is of great importance in a wide range of application areas in science and engineering. It represents major challenges because of the nonlinear potential. The main aim of this chapter is to present symmetric and arbitrarily high-order time-stepping integrators and analyse their stability, convergence and long-time behaviour for the semilinear Klein–Gordon equation. To achieve this, under the assumption of periodic boundary conditions, an abstract ordinary differential equation (ODE) and its operator-variation-of-constants formula are formulated on a suitable function space based on operator spectrum theory. By applying a two-point Hermite–Birkhoff interpolation to the nonlinear integrals that appear in the operator-variation-of-constants formula, as a result, a suitable spatial discretisation leads to the fully discrete scheme, which needs only a weak temporal smoothness assumption.

Suggested Citation

  • Xinyuan Wu & Bin Wang, 2021. "High-Order Symmetric Hermite–Birkhoff Time Integrators for Semilinear KG Equations," Springer Books, in: Geometric Integrators for Differential Equations with Highly Oscillatory Solutions, chapter 0, pages 299-349, Springer.
    • Handle: RePEc:spr:sprchp:978-981-16-0147-7_10
    DOI: 10.1007/978-981-16-0147-7_10
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