A modified explicit numerical scheme for the two-dimensional sine-Gordon equation

Abstract

A fourth-order rational approximant to the matrix-exponential term in a three-time-level recurrence relation is used to transform the two-dimensional sine-Gordon equation into a second-order initial-value problem. The resulting nonlinear system is solved using an appropriate predictor–corrector (P-C) scheme in which the predictor is an explicit one of second order. The procedure of the corrector is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the nonlinear method and the predictor–corrector are analysed for local truncation error and stability. The MPC scheme has been tested on line and circular ring solitons known from the literature, and numerical experiments have proved that there is an improvement in accuracy over the standard predictor–corrector implementation.

Keywords:

• Soliton
• Sine-Gordon
• Finite-difference method
• Predictor–corrector

Acknowledgements

This research was co-funded 75% by the European Union and 25% by the Greek Government under the framework of the Education and Initial Vocational Training Program—Archimedes, Technological Educational Institution (T.E.I.) Athens project ‘Computational Methods for Applied Technological Problems’.

Additional information

Notes on contributors

A.G. Bratsos

Email: [email protected]