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.

为解决非均匀张量积网格上的Schrödinger方程，开发了两种有限差分时域（FDTD）方法。第一个是将均匀网格上的标准二阶精确空间差分方案扩展为非均匀网格，而第二个是使用扩展模板来使用更高阶的精确空间差分方案。基于离散时间稳定性理论，推导了两种方案的时间步长的上限。结果表明，与已知的稳定性标准相比，以这种方式得出的时间步长更大。此外，研究数值色散误差与时间步长，空间步长和传播方向的关系。