Padé numerical schemes for the sine-Gordon equation

Fractal structure of the soliton scattering for the graphene superlattice equation

“…The most efficient method in the comparison of implicit Padé numerical schemes for the sine-Gordon equation in Ref. [36] has been selected; concretely, a fourth-order in space generalization of Strauss-Vázquez method, which is the second-order in time and uses a treatment of the nonlinearity that ensures good energy conservation. The collision of a kink and an antikink with the same, but opposite speed is studied under periodic boundary conditions, so an imbricated initial condition is used.…”

“…The analysis of the kink-antikink collisions in the GSLeq requires the use of an efficient method due to the large amount of numerical simulations needed. The authors have recently compared eight implicit finite difference methods with Padé approximations in space for the sine-Gordon equation (sGeq), without [36] and with [37] Richardson extrapolation; these methods were inspired in the conservative methods developed by Strauss and Vázquez [38] and Guo Ben-Yu et al [39]. Our comparison among these Padé methods indicates that the most efficient method for small global error in terms of accuracy and computational cost is a (4,0)-Padé method based on the Guo Ben-Yu et al method without Richardson extrapolation [37].…”

Padé schemes with Richardson extrapolation for the sine-Gordon equation

“…Recently, we have developed (0, 4), (2,2), (2,4), and (4, 4) Padé methods for the sGE using the nonlinearity treatment developed by Strauss and Vázquez [10]; our results show that the most cost-effective ones are those of higher order, being the spatial order of accuracy more relevant for accuracy than the energy conservation property, even in long-time integrations [20]. Similar conclusions have been obtained for other schemes in older studies [13,21,22].…”