High-order compact ADI (HOC-ADI) method for solving unsteady 2D Schrödinger equation

In this paper, a high-order compact (HOC) alternating direction implicit (ADI) method is proposed for the solution of the unsteady two-dimensional Schrödinger equation. The present method uses the fourth-order Padé compact difference approximation for the spatial discretization and the Crank-Nicolson scheme for the temporal discretization. The proposed HOC-ADI method has fourth-order accuracy in space and second-order accuracy in time. The resulting scheme in each ADI computation step corresponds to a tridiagonal system which can be solved by using the one-dimensional tridiagonal algorithm with a considerable saving in computing time. Numerical experiments are conducted to demonstrate its efficiency and accuracy and to compare it with analytic solutions and numerical results established by some other methods in the literature. The results show that the present HOC-ADI scheme gives highly accurate results with much better computational efficiency.