An efficient numerical treatment for the asymptotic behaviour of the nonlinear Airy-type problems

Abstract

This study focuses on symplectic integrators for numerical evaluation of the asymptotic solutions of the nonlinear Airy-type equations obtained by reducing the nonlinear dispersive equations. Since the nature of Airy-type equations has both highly oscillatory slow decay and exponential fast decay, most of classical integrators are not able to correctly exhibit challenging physical behaviour. We use specially designed symplectic integrators combining splitting methods with Magnus integrators to catch asymptotic behaviour of nonlinear Airy-type equations efficiently, even for large step sizes. Efficiency of the proposed methods for given problems is discussed. Moreover, numerical results obtained by the proposed methods are compared with the existing results in the literature. © 2020 Elsevier B.V.