A modified predictor-corrector scheme for the Klein–Gordon equation

Abstract

A numerical method based on a three-time level finite-difference scheme has been proposed for the solution of the two forms of the Klein–Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor-corrector scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector is modified by considering, as known, the already evaluated corrected values instead of the predictor ones. This modified scheme is applied to problems possessing periodic, kinks and soliton waves. The accuracy as well as the long-time behaviour of the proposed scheme is discussed and comparisons with the relevant known in the bibliography schemes are given.

Keywords:

• soliton
• Klein–Gordon equation
• finite-difference method
• predictor-corrector

2000 AMS Subject Classifications :

• 35Q51
• 81Q05
• 65M06
• 78M20
• 65Y10

Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions, which have improved the paper.

Notes

^†Dedicated to Professor E.H. Twizell