High-order implicit Galerkin-Legendre spectral method for the   two-dimensional Schrodinger equation
  In this paper, we propose Galerkin-Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrodinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a Galerkin-Legendre spectral method for discretizing spatial derivatives, and then employ the implicit Runge-Kutta method for the time integration of the resulting linear first-order system of ordinary differential equations in complex domain. We derive the spectral rate of convergence for the proposed method in the L^2-norm for the semidiscrete formulation.   Numerical experiments show our formulation have high-order accurate, and have the exponential rates of convergence in space. 
Wenjie Liu and Boying Wu
math.NA