Determination of the direction that has maximum phase-velocity for the 2-D and 3-D FDTD methods based on Yee´s algorithm

To easily evaluate the performance of the numerical dispersion of the FDTD method, a simple and efficient manner to theoretically determine the special direction that has the maximum phase-velocity for arbitrarily shaped two-dimensional (2-D) and three-dimensional (3-D) Yee´s cells is proposed. It is found that the wave propagating along the special direction obeys the following physical rule: the time that the wave takes to traverse the spatial step sizes along each of the main axes of the cell is identical. It is demonstrated that this special direction is closely related to the aspect ratio(s) of the cells, rather than the diagonals of the cells. Moreover, the mathematical expressions for the maximum phase-velocity of the 2-D and 3-D cells with arbitrary aspect ratios are derived for the first time.