An efficient second-order finite difference method for the one-dimensional Schrödinger equation with absorbing boundary conditions

Abstract

A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the one-dimensional Schrödinger equation. Our approximation is based on the Padé expansion of the square root function in the complex plane. By introducing a constant damping term to the governing equation and modifying the standard Crank--Nicolson implicit scheme, we show that the fully discrete numerical scheme is unconditionally stable if the order of Padé expansion is chosen from our criterion. In this case, an optimal-order asymptotic error estimate is proved for the numerical solutions. Numerical examples are provided to support the theoretical analysis and illustrate the performance of the proposed numerical scheme.