Guilin Sun

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 Maxwells 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.

Analysis and numerical experiments on the numerical dispersion of two-dimensional ADI-FDTD

The numerical dispersion relations in the literature are inconsistent for the alternate-direction-implicit finite-difference time-domain (ADI-FDTD) method. By analysis of the amplification factors, the numerical dispersion relation is rederived and verified with numerical experiments, with good agreement. The inconsistency of the numerical dispersion relation is resolved. It is shown that ADI-FDTD has some fundamental limits. For a given time step size, there is a velocity error even for zero spatial mesh. For a given spatial mesh size, the mesh does not support a numerical wave at certain time step sizes. As the Nyquist sampling limit is approached, the velocity of the wave approaches zero. At about twice the Nyquist limit, the wave does not propagate. Hence, the Nyquist criterion should be respected in choosing the time step size.

Quantification of the truncation errors in finite-difference time-domain methods

To characterize a finite-difference time-domain (FDTD) scheme, the truncation error using Taylors series and the numerical dispersion are often used. Truncation error analysis determines the order of accuracy, but cannot differentiate one scheme from another if they have the same order of accuracy. The theoretical relation for the numerical dispersion sometimes may be difficult to obtain. This paper introduces another way to characterize the error of an FDTD scheme quantitatively. This is the truncation error with plane wave propagation. The analytical expressions for such truncation errors for Yee s FDTD and Crank-Nicolson FDTD are derived, and the errors are compared.

Corrections to "Efficient Implementations of the Crank–Nicolson Scheme for the Finite-Difference Time-Domain Method

The authors correct a misprint in an equation in their original article (see ibid., vol.54, no.5, p.2275-84, May 2006).

Unconditionally stable solution for wave equations of the second order

This paper reports an unconditionally stable and implicit method to solve the electromagnetic wave equations of the second order numerically without the Courant limit. By using the Crank-Nicolson (CN) scheme with the Douglas-Gunn (DG) algorithm, the numerical solution is second order accurate in both space and time. Two schemes are introduced and compared, and the numerical dispersion relations are given. Numerical experiments agree with the numerical dispersion predicted in theory.

The unconditionally-stable cycle-sweep method for 3D FDTD

To overcome the Courant limit in the explicit finite-difference time-domain (FDTD) methods, several unconditionally-stable methods have been proposed. This paper presents the cycle-sweep Crank-Nicolson method for solving 3D Maxwells Equations. The cycle-sweep method is unconditionally stable and 2nd order accurate in both time and space, and solves only six tridiagonal matrices instead of nine in the approximate-factorization-splitting method we reported previously. The cycle-sweep method is much more efficient than either a direct implementation of the Crank-Nicolson scheme or the approximate-factorization-splitting method.