Unconditionally-stable FDTD method based on Crank-Nicolson scheme for solving three-dimensional Maxwell equations

The approximate-factorisation-splitting (CNAFS) method as an efficient implementation of the Crank-Nicolson scheme for solving the three-dimensional Maxwell equations in the time domain, using much less CPU time and memory than a direct implementation, is presented. At each time step, the CNAFS method solves tridiagonal matrices successively instead of solving a huge sparse matrix. It is shown that CNAFS is unconditionally stable and has much smaller anisotropy than the alternating-direction implicit (ADI) method, though the numerical dispersion is the same as in the ADI method along the axes. In addition, for a given mesh density, there will be one value of the Courant number at which the CNAFS method has zero anisotropy, whereas the CrankNicolson scheme always has anisotropy. Analysis shows that both ADI and CNAFS have time step-size limits to avoid numerical attenuation, although both are still unconditionally stable beyond their limit.