High-order implicit Galerkin–Legendre spectral method for the two-dimensional Schrödinger equation

In this paper, we propose Galerkin–Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrödinger 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 L²-norm for the semidiscrete formulation. Numerical experiments show our formulation have high-order accuracy.