7. FDTD in Cartesian and Spherical Grids

The numerical dispersion relation is derived for the finite-difference time-domain method when implemented on spherical grids using Maxwell’s equations in spherical coordinates. Derivation is appropriately based on elementary spherical functions which renders the resulting numerical dispersion relation valid for all spherical FDTD space including near the singular regions at the origin and along the z-axis. Accuracy of this relation is verified through convergence tests to the continuous-space limit and the Cartesian FDTD limit far from the origin. Numerical dispersion analyses are carried out to demonstrate numerical wavenumber error bounds and their dependence on absolute position as well as on spherical solutions’ modes. The chapter is concluded by visiting the existing challenges of designing absorbing boundary conditions for spherical FDTD when the grid truncation is in the near vicinity of the origin. Such a design challenge can be effectively studied in the future with the aid of the derived spherical FDTD numerical dispersion relation.