Abstract
Alternating direction implicit (ADI) schemes are proposed for the solution of the two-dimensional coupled nonlinear Schrödinger equation. These schemes are of second- and fourth-order accuracy in space and second order in time. The resulting schemes in each ADI computation step correspond to a block tridiagonal system which can be solved by using one-dimensional block tridiagonal algorithm with a considerable saving in computational time. These schemes are very well suited for parallel implementation on a high performance system with many processors due to the nature of the computation that involves solving the same block tridiagonal systems with many right hand sides. Numerical experiments on one processor system are conducted to demonstrate the efficiency and accuracy of these schemes by comparing them with the analytic solutions. The results show that the proposed schemes give highly accurate results.
Highlights
In this paper, we consider the coupled nonlinear Schrodinger equation iψt + δ + (ψ2 + αφ2) ψ = 0,(x, y, t) ∈ Ω × (0, T], (1) iφt + δ + (αψ2 + φ2) φ = 0 with initial conditions ψ (x, y, 0) = f1 (x, y), φ (x, y, 0) = f2 (x, y), (2)(x, y) ∈ Ω and Dirichlet boundary conditions ψ (x, y, t) = g1 (x, y, t), φ (x, y, t) = g2 (x, y, t), (3)(x, y, t) ∈ ∂Ω ×
To derive the numerical schemes for solving system (1), we consider the domain of interest Ω = [a, b] × [a, b], such that (x, y) ∈ Ω
By introducing a new intermediate vector U∗l,m, we propose a D’Yakonov [12, 17] Alternating direction implicit (ADI)-like scheme for the coupled system
Summary
(x, y) ∈ Ω and Dirichlet boundary conditions ψ (x, y, t) = g1 (x, y, t) , φ (x, y, t) = g2 (x, y, t) , (3). We assume that Ω = [a, b] × [c, d], ∂Ω is the boundary of Ω, (0, T] is the time interval, and f1(x, y), f2(x, y), g1(x, y, t), and g2(x, y, t) are given sufficiently smooth functions. Many numerical methods have been developed for solving the coupled nonlinear Schrodinger [2,3,4,5,6]. Many published works for solving the two-dimensional nonlinear Schrodinger equation are given in [7,8,9,10,11]. We are going to derive an ADI method for solving the two-dimensional coupled nonlinear Schrodinger equation
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
