Extended study of Poynting theorem and reciprocity on nonuniform FDTD meshes

The physical mechanisms behind numerical errors arising on nonuniform FDTD meshes are revealed and evaluated, focusing on the basic step-in-cell-size mesh nonuniformity and standard as well as compensated FDTD algorithms. Beginning with rigorous derivations of mathematical formulas for the numerical reflection coefficient, it is shown that wave impedance for the travelling wave and phase shift between the standing wave E- and H-fields are conserved. However, field discontinuity across the mesh discontinuity results thereupon. As a consequence the Poynting vector increases when the wave travels from fine to coarse mesh, and decreases in the opposite case. The Poynting theorem and reciprocity condition are violated. In standard FDTD, these numerical artifacts are at the level of -32 dB for the benchmarking case of ten coarse cells per wavelength and 10:1 mesh grading ratio. The compensated FDTD scheme suppresses the nonphysical effects to the level of -60 dB, which is practically undetectable in FDTD analyses of real problems.