  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.
