A linearized, decoupled, and energy‐preserving compact finite difference scheme for the coupled nonlinear Schrödinger equations. Issue 3 (17th November 2016)

• Record Type: Journal Article
• Authors: Wang, Tingchun
• Is Part Of: Numerical methods for partial differential equations. Volume 33:Issue 3(2017:May)
• Page Start: 840
• Page End: 867
• Publication Date: 2016-11-17
• Subjects: coupled nonlinear Schrödinger equations -- linearized and decoupled scheme -- compact finite difference scheme -- mass and energy conservation -- unconditional convergence -- optimal point‐wise error estimate; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22125
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract : In this article, a decoupled and linearized compact finite difference scheme is proposed for solving the coupled nonlinear Schrödinger equations. The new scheme is proved to preserve the total mass and energy which are defined by using a recursion relationship. Besides the standard energy method, an induction argument together with an H 1 technique are introduced to establish the optimal point‐wise error estimate of the proposed scheme. Without imposing any constraints on the grid ratios, the convergence order of the numerical solution is proved to be of O ( h 4 + τ 2 ) with mesh size h and time step τ . Numerical results are reported to verify the theoretical analysis, and collision of two solitary waves are also simulated. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 840–867, 2017