Accuracy of Three Unconditionally-Stable FDTD Schemes for Solving Maxwell's Equations

Abstract

This paper discusses accuracy limitations due to numerical dispersion and time step size for three implicit unconditionally- stable FDTD methods: Alternate-Direction- Implicit (ADI), Crank-Nicolson (CN) and Crank-Nicolson- Douglas-Gunn (CNDG). It is shown that for a uniform mesh, the three methods have the same numerical phase velocity along the axes, but have large differences along the diagonals. The ADI method has two orders-of-magnitude larger anisotropy than that of CN and CNDG. CNDG has no anisotropy at certain Courant numbers and mesh densities. At the limit of zero spatial mesh size, the three methods have different "intrinsic temporal dispersion" for a given time step size: CN has no anisotropy; ADI has positive anisotropy and CNDG has negative anisotropy, which is much smaller than ADI. The Nyquist sampling theorem provides a fundamental upper bound on the time step size for all three methods. It is shown that for ADI and CN the practical upper bound is close to the Nyquist limit, but for CNDG is half the Nyquist limit.