A numerical method for the one-dimensional Sine-Gordon equation

Abstract

A numerical method based on a predictor-corrector (P-C) scheme arising from the use of rational approximants of order 3 to the matrix-exponential term in a three-time level recurrence relation is applied successfully to the one-dimensional sine-Gordon equation, already known from the bibliography. In this P-C scheme a modification in the corrector (MPC) has been proposed according to which the already evaluated corrected values are considered. The method, which uses as predictor an explicit finite-difference scheme arising from the second order rational approximant and as corrector an implicit one, has been tested numerically on the single and the soliton doublets. Both the predictor and the corrector schemes are analyzed for local truncation error and stability. From the investigation of the numerical results and the comparison of them with other ones known from the bibliography it has been derived that the proposed P-C/MPC schemes at least coincide in terms of accuracy with them. © 2007 Wiley Periodicals, Inc.