Application of high-order SF-SFDTD scheme to solving a time-dependent Schrödinger equation

Abstract

The traditional finite-difference time-domain (FDTD (2, 2)) method with second-order numerical accuracy in time and space has been extensively employed in the field of quantum mechanics to solve the Schrödinger equation. Nevertheless, the presence of the Courant-Friedrichs-Lewy (CFL) condition imposes limitations on the grid size in the computational space, thereby constraining the admissible range of time steps. Accordingly, the efficiency of the FDTD(2, 2) method significantly decreases. In addition, the second-order numerical accuracy of the FDTD(2, 2) method both in time domain and in space domain often results in significant error accumulation during calculations, thereby undermining the fidelity of the simulation results. To surmount the constraints imposed by the CFL stability conditions and enhance the accuracy of computations, a novel approach termed SF-SFDTD(3, 4) method has been proposed, with 3 and 4 referring to the accuracy in space and time, respectively. This method combines spatial filtering (SF) with the high-order symplectic finite-difference time-domain (SFDTD) method. Its primary objective is to solve the time-dependent Schrödinger equation while ensuring time stability and scalability. The SF-SFDTD(3, 4) method obviates the need for further deriving the iterative formula employed in the conventional SFDTD(3, 4) method. Therefore, the method under consideration exhibits a remarkable degree of compatibility with its traditional counterpart. It is merely necessary to include a spatial filtering operation during each numerical iteration to eliminate spatial high-frequency components arising from the utilization of time step sizes that fail to satisfy the CFL stability condition, thereby ensuring the stability of the numerical scheme. Moreover, when the time step value satisfies the CFL stability condition, the amplitude of the high-frequency component approaches zero, thereby exerting a minimal influence on the accuracy of the computational results. The adoption of time steps that do not meet the CFL stability conditions leads to an amplification in the amplitude of the high-frequency component. However, this finding solely affects the stability of the computational results, and the elimination of these unstable high-frequency components scarcely affect the accuracy of the computational results. The SF-SFDTD(3, 4) retains the simplicity and efficacy inherent in the traditional SFDTD(3, 4) methods, while enhancing computational efficiency. Additionally, the numerical stability and dispersion error of the SF-SFDTD(3, 4) method are analyzed theoretically. Finally, the validity and efficacy of the proposed method are corroborated through numerical illustrations.