Two-grid discretization schemes for nonlinear Schrödinger equations

Abstract

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of twocoupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction-diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.

A Two-Grid Finite-Element Method for the Nonlinear Schrödinger Equation

“…Chang et al [3] combined the two-grid discretization together with the predictor-corrector method and developed an algorithm for computing the extremum eigenpairs of the discrete Schrödinger eigenvalue problem. Chien et al [6] proposed two-grid discretization schemes with two-loop continuation algorithms for nonlinear Schrödinger equations, where the centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs. Numerical experiments have shown that these schemes were efficient, but no rigorous error analysis were given.…”

A Two-Grid Finite-Volume Method for the Schrödinger Equation

“…On the other hand, the two-grid discretization method, proposed originally by Xu [20] in 1992, is an efficient numerical method. And it was further investigated and applied to solving many problems, such as nonlinear parabolic equations [21], nonlinear elasticity problems [22], nonlinear PDEs [23], Navier-Stokes equations [24,25], evolution equations [26], two-phase mixed-domain PEMFC model [27], nonlinear natural convection system [28], Schrödinger equations [1,[29][30][31][32][33][34] and so on.…”

Implicit–explicit multistep methods for general two-dimensional nonlinear Schrödinger equations

“…In nonlinear optics, the CNLS equations can be used to model the propagation of the multimode soliton pulses at least in two channels simultaneously [31], where the multichannel bit-parallel wavelength optical fiber networks are considered to increase the transmission capacity of the lightwave systems [39]. There are numerous investigations on numerical solutions of CNLS equations including finite difference method [20], meshless method [11] and two-grid method [10].…”