Stability and error analysis for a second-order fast approximation of the one-dimensional schrödinger equation under absorbing boundary conditions

Abstract

A second-order Crank-Nicolson finite difference method, integrating a fast approximation of an exact discrete absorbing boundary condition, is proposed for solving the one-dimensional Schrödinger equation in the whole space. The fast approximation is based on Gaussian quadrature approximation of the convolution coefficients in the discrete absorbing boundary conditions. It approximates the time convolution in the discrete absorbing boundary conditions by a system of O(log 2 N) ordinary differential equations at each time step, where N denotes the total number of time steps. Stability and an error estimate are presented for the numerical solutions given by the proposed fast algorithm. Numerical experiments are provided, which agree with the theoretical results and show the performance of the proposed numerical method.