A fast BDF2 Galerkin finite element method for the one-dimensional time-dependent Schrodinger equation with artificial boundary conditions

In this paper, we propose an efficient numerical scheme with linear complexity for the one-dimensional time-dependent Schrodinger equation on unbounded domains. The artificial boundary method is used to address the unboundedness of the domain. By applying the two-step backward difference formula for time discretization and performing the Z- transform, we derive an exact semi-discrete artificial boundary condition of the Dirichlet-to-Neumann type. To expedite the discrete temporal convolution involved in the exact semi-discrete artificial boundary conditions, we design a fast algorithm based on the best relative Chebyshev approximation of the square-root function. The Galerkin finite element method is used for spatial discretization. By introducing a constant damping term to the original Schrodinger equation, we present a complete error estimate for the fully discrete problem. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed numerical scheme. (c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.