A Galerkin Splitting Symplectic Method for the Two Dimensional Nonlinear Schrödinger Equation (Advances in Applied Mathematics and Mechanics, 2018)
In this paper, we propose a Galerkin splitting symplectic (GSS) method forsolving the 2D nonlinear Schrödinger equation based on the weak formulation of the equation. First, the model equation is discretized by the Galerkin method in spatialdirection by a selected finite element method and the semi-discrete system is rewrittenas a finite-dimensional canonical Hamiltonian system. Then the resulted Hamiltoniansystem is split into a linear Hamiltonian subsystem and a nonlinear subsystem. Thelinear Hamiltonian subsystem is solved by the implicit midpoint method and the nonlinearsubsystem is integrated exactly. By the Strang splitting method, we obtain afully implicit scheme for the 2D nonlinear Schrödinger equation (NLS), which is symmetric and of order 2 in time. Furthermore, we apply the FFT technique to improvecomputation efficiency of the new scheme. It is proven that our scheme preserves themass conservation and the symplectic conservation. Comprehensive numerical experimentsare carried out to illustrate the accuracy of the scheme as well as its conservativeproperties.