The stability and numerical dispersion analyses of high-order symplectic FDTD scheme for solving time-dependent Schrödinger equation

A high-order symplectic FDTD (SFDTD) framework for solving the time-dependent Schrödinger equation is established. The third-order symplectic integrators and fourth-order collocated differences are employed in the time and space domains, respectively. The stabilities and numerical dispersions of FDTD(2,2), higher-order FDTD(2,4), and SFDTD(3,4) schemes are analyzed. We found that the stability limit of the SFDTD(3,4) scheme can be larger than that of the traditional FDTD(2,2) method through careful optimization of symplectic integrators. Moreover, the SFDTD(3,4) scheme and the FDTD(2,4) approach show better numerical dispersions than the traditional FDTD(2,2) method.