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Original Articles

A fourth order numerical scheme for the one-dimensional sine-Gordon equation

Pages 1083-1095 | Received 16 Mar 2007, Accepted 22 May 2007, Published online: 04 Mar 2011
 

Abstract

A numerical scheme arising from the use of a fourth order rational approximants to the matrix-exponential term in a three-time level recurrence relation is proposed for the numerical solution of the one-dimensional sine-Gordon (SG) equation already known from the bibliography. The method for its implementation uses a predictor–corrector scheme in which the corrector is accelerated by using the already evaluated corrected values modified predictor–corrector scheme. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. Both the predictor and the corrector schemes are analysed for stability. The predictor–corrector/modified predictor–corrector (P-C/MPC) schemes are tested on single and soliton doublets as well as on the collision of breathers and a comparison of the numerical results with the corresponding ones in the bibliography is made. Finally, conclusions for the behaviour of the introduced MPC over the standard P-C scheme are derived.

2000 AMS Subject Classification :

Acknowledgements

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

Additional information

Notes on contributors

A. G. Bratsos

Email: [email protected]

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