Padé numerical schemes with Richardson extrapolation for the sine–Gordon equation

Abstract

Four novel implicit finite difference methods with ( q + s ) -th order in space based on ( q, s )-Padé approximations have been analyzed and developed for the sine-Gordon equation. Specifically, (4,0)-, (2,2)-, (4,2)-, and (4,4)-Padé methods. All of them share the treatment for the nonlinearity and integration in time, specifically, the one that results in an energy-conserving (2,0)-Padé scheme. The five methods have been developed with and without Richardson extrapolation in time. All the methods are linearly, unconditionally stable. A comparison among them for both the kink–antikink and breather solutions in terms of global error, computational cost and energy conservation is presented. Our results indicate that the (4,0)- and (4,4)-Padé methods without Richardson extrapolation are the most cost-effective ones for small and large global error, respectively; and the (4,4)-Padé methods in all the cases when Richardson extrapolation is used.