Numerical stability and error analysis of transformation optics for electromagnetic simulation in time-domain

Applying transformation optics (TO) for converting non-uniform physical grids into uniform numerical grids with an additional derived anisotropic medium and an analysis of numerical stability and error in an electromagnetic finite-difference time-domain (FDTD) simulation are presented. Traditionally, the maximum time-step is limited by the smallest cell size. However, due to the invariance of time, when smaller physical grids are converted into a numerical system, the phase velocity of light can be greater than that in free space. The conventional Fourier analysis of numerical dispersion to seek numerical stability is no longer applied, but rather there is further consideration of the properties of the derived medium. Thus, we derive the numerical dispersion relation to seek a more general numerical stability condition for a converted numerical system. It is found to be consistent with that of a non-uniform physical system. Also, an error analysis is performed by comparing the differences in the electric fields simulated in the converted numerical system with those in the uniform physical system. We show that they are almost of the same order; that is, there is almost no extra error induced when applying the proposed numerical simulation method.