LAUSR

A closed-form expression is derived for the numerical dispersion relation of the 2-D locally one-dimensional finite-difference time-domain (LOD-FDTD) method in lossy media. In contrast to the lossless formulation, we found that transverse-electric (TE$_{z}$) and transverse-magnetic (TM$_{z}$) waves in lossy media exhibit different numerical dispersion relations. Moreover, when the material relaxation-time constant is not well resolved by the integration time-step, the TM$_{z}$ case shows much worse accuracy than the TE$_{z}$ case. To remove this limitation, a split-field LOD-FDTD formulation for TM $_{z}$ waves is then considered, which exhibits the same dispersion relation as the LOD-FDTD method for TE$_{z}$ waves. The validity of the theoretical results is illustrated through numerical simulations.