Efficient implementations of the Crank-Nicolson scheme for the finite-difference time-domain method

When a finite-difference time-domain (FDTD) method is constructed by applying the Crank-Nicolson (CN) scheme to discretize Maxwell's equations, a huge sparse irreducible matrix results, which cannot be solved efficiently. This paper proposes a factorization-splitting scheme using two substeps to decompose the generalized CN matrix into two simple matrices with the terms not factored confined to one sub-step. Two unconditionally stable methods are developed: one has the same numerical dispersion relation as the alternating-direction implicit FDTD method, and the other has a much more isotropic numerical velocity. The limit on the time-step size to avoid numerical attenuation is investigated, and is shown to be below the Nyquist sampling rate. The intrinsic temporal numerical dispersion is discussed, which is the fundamental accuracy limit of the methods.

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