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A fourth order numerical scheme for the one-dimensional sine-Gordon equation. (English) Zbl 1145.65053

Summary: 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 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.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
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