An improved numerical scheme for the sine‐Gordon equation in 2+1 dimensions

A rational approximant of order 4, which is applied to a three‐time‐level recurrence relation, is used to transform the initial/boundary‐value problem associated with the two‐dimensional sine‐Gordon (SG) equation arising in the Josephson junctions problem. The resulting non‐linear system, which is analyzed for stability, is solved using an appropriate predictor–corrector (P–C) scheme, in which an explicit scheme of order 2 is used as predictor. For the implementation of the corrector, in order to avoid extended matrix evaluations, an auxiliary vector was successfully introduced. In this P–C scheme, a modification in the corrector has been proposed according to which the already evaluated corrected values are considered. The behavior of this P–C scheme is tested numerically on line and ring solitons known from the bibliography regarding the SG equation and conclusions for both undamped and damped problems are derived. Copyright © 2008 John Wiley & Sons, Ltd.