Nonuniform and higher-order FDTD methods for the Schrödinger equation

Two Finite-Difference Time-Domain (FDTD) methods are developed for solving the Schrödinger equation on nonuniform tensor-product grids. The first is an extension of the standard second-order accurate spatial differencing scheme on uniform grids to nonuniform grids, whereas the second utilizes a higher-order accurate spatial scheme using an extended stencil. Based on discrete-time stability theory, an upper bound is derived for the time step of both proposed schemes. It is shown that the time step derived in this way can be larger compared to the known stability criterion. Furthermore, the numerical dispersion error is investigated as a function of the time step, the spatial step and the propagation direction. Numerical experiments are compared with analytical solutions and demonstrate the increased accuracy of the higher-order scheme as well as the advantageous properties of nonuniform gridding.


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