Padé numerical schemes for the sine-Gordon equation

• Published in: Applied mathematics and computation Vol. 358; pp. 232 - 243
• Main Authors: Martin-Vergara, Francisca, Rus, Francisco, Villatoro, Francisco R.
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2019
• Subjects: Implicit time integration; Padé numerical methods; Sine-Gordon equation; Solitons; Implicit time integration; Sine-Gordon equation; Padé numerical methods; Solitons
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2019.04.042

•Four novel, high-order, Padé numerical methods are developed for the sine-Gordon equation.•High-order methods are more cost-effective than low-order energy-conserving ones.•In long-time integration, some recommendations of the selection of the parameters of the method are given. 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) Padé 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) Padé methods; additionally, the energy conserving, Strauss–Vázquez scheme has been considered in a (0,2) Padé 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) Padé 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.