Partially Implicit FDTD (PI-FDTD) Method for Lower Dispersion and Anisotropic Errors

Abstract

A new partially implicit 3D FDTD (PI-FDTD) method is introduced. The method is not unconditionally stable, but its stability limit is 6 times higher than that of the classic Yee&#x2019;s FDTD method. The method has only one implicit update equation and uses four split time steps. A study has been carried out to compare the performances of the new method and the ADI-FDTD, LOD-FDTD, and HIE-FDTD methods. ADI-FDTD and LOD-FDTD methods are unconditionally stable and the HIE-FDTD method is conditionally stable with a stability limit of 3.46 times above the Yee&#x2019;s FDTD method. The dispersion and anisotropic errors of the PI-FDTD method are considerably lower than that of these methods and for small cell sizes, the new method is almost isotropic. For example at the stability limit of the PI-FDTD method with a uniform cell size of <inline-formula> <tex-math notation="LaTeX">$\lambda \mathord {\left /{ {{ {50}}} }\right. }$ </tex-math></inline-formula> the maximum dispersion errors and anisotropic errors are 0.15&#x0025; and 0.031&#x0025; for the new method and 1.66&#x0025; and 0.64&#x0025; for the ADI-FDTD and LOD-FDTD methods. At the stability limit of the HIE-FDTD method, the maximum dispersion errors and anisotropic errors of the PI-FDTD method are also less than HIE-FDTD, ADI-FDTD, and LOD-FDTD methods. The differences in errors are more pronounced for larger cell sizes. Although the new method uses a four time split step scheme and the ADI-FDTD and LOD-FDTD methods use a two time split step scheme, the computational time of the PI-FDTD method is less, as it uses an implicit scheme for only one of the field components when updating. The computational time of the PI-FDTD method is the same as that of the HIE-FDTD method. The study has revealed that the new method can be used over much wider frequency bandwidths.