Approximate Crank-Nicolson schemes for the 2-D finite-difference time-domain method for TEz waves

Two implicit finite-difference time-domain (FDTD) methods are presented in this paper for a two-dimensional TE_z wave, which are based on the unconditionally-stable Crank-Nicolson scheme. To treat PEC boundaries efficiently, the methods deal with the electric field components rather than the magnetic field. The "approximate-decoupling method" solves two tridiagonal matrices and computes only one explicit equation for a full update cycle. It has the same numerical dispersion relation as the ADI-FDTD method. The "cycle-sweep method" solves two tridiagonal matrices, and computes two equations explicitly for a full update cycle. It has the same numerical dispersion relation as the previously-reported Crank-Nicolson-Douglas-Gunn algorithm, which solves for the magnetic field. The cycle-sweep method has much smaller numerical anisotropy than the approximate-decoupling method, though the dispersion error is the same along the axes as, and larger along the 45° diagonal than ADI-FDTD. With different formulations, two algorithms for the approximate-decoupling method and four algorithms for the cycle-sweep method are presented. All the six algorithms are strictly nondissipative, unconditionally stable, and are tested by numerical computation in this paper. The numerical dispersion relations are validated by numerical experiments, and very good agreement between the experiments and the theoretical predication is obtained.