Analysis of the Numerical Error Caused by the Stair-Stepped Approximation of a Conducting Boundary in FDTD Simulations of Electromagnetic Phenomena

A rigorous analysis of the numerical error associated with the use of stair-stepped (saw-tooth) approximation of a conducting boundary for finite-difference time-domain (FDTD) simulations is presented. First, a dispersion analysis in two dimensions is performed to obtain the numerical reflection coefficient for a plane wave scattered by a perfectly conducting wall, tilted with respect to the axes of the finite-difference grid, under both transverse electric and transverse magnetic polarizations. The characteristic equation for surface waves that can be supported by such saw-tooth conducting surfaces is derived. This equation leads naturally to expressions that show the dependence of the propagation constant along the boundary and the attenuation constant perpendicular to it on cell size and wavelength. Finally, numerical simulations are presented which demonstrate the effects predicted by the dispersion analysis.