Pade numerical schemes for the sine-Gordon equation

The sine-Gordon equation turn up in several problems in science and engineering. Although it is integrable, in practical applications, its numerical solution is powerful and versatile. Four novel implicit finite difference methods based on (q, s) Pade approximations with (q + s)th order in space have been developed and analyzed for this equation; all share the same treatment for the nonlinearity and integration in time. Concretely, (0,4), (2,2), (2,4), and (4,4) Pade methods; additionally, the energy conserving, Strauss-Vazquez scheme has been considered in a (0,2) Pade implementation. These methods have been compared among them for both the kink-antikink and breather solutions in terms of global error, computational cost and energy conservation. The (0,4) and (2,4) Pade methods are the most cost-effective ones for small and large global error, respectively. Our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations. (C) 2019 Elsevier Inc. All rights reserved.