Detailed comparison of numerical methods for the perturbed sine-Gordon equation with impulsive forcing
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The properties of various numerical methods for the study of the perturbed sine-Gordon (sG) equation with impulsive forcing are investigated. In particular, finite difference and pseudo-spectral methods for discretizing the equation are considered. Different methods of discretizing the Dirac delta are discussed. Various combinations of these methods are then used to model the soliton–defect interaction. A comprehensive study of convergence of all these combinations is presented. Detailed explanations are provided of various numerical issues that should be carefully considered when the sG equation with impulsive forcing is solved numerically. The properties of each method depend heavily on the specific representation chosen for the Dirac delta—and vice versa. Useful comparisons are provided that can be used for the design of the numerical scheme to study the singularly perturbed sG equation. Some interesting results are found. For example, the Gaussian approximation yields the worst results, while the domain decomposition method yields the best results, for both finite difference and spectral methods. These findings are corroborated by extensive numerical simulations.
KeywordsFinite difference methods Impulsive forcing Sine-Gordon equation Spectral methods
We thank Roy Goodman, Panayotis Kevrekidis, and Jingbo Xia for many valuable discussions. This study was partially supported by the National Science Foundation under Award Number DMS-0908399.
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