Convolutional CFS-PML for the 2-D Crank–Nicolson FDTD Scheme and Its Application in Simulation of Ultralow Frequency Electromagnetic Problems

A convolutional implementation of the perfectly matched layer (PML) absorbing boundary condition (ABC) with complex frequency-shifted (CFS) constitutive parameters is developed for the original 2-D Crank-Nicolson (CN) scheme of the finite-difference time-domain (FDTD) method. The proposed CN-FDTD method leverages the benefits of both CFS-PML and the unconditionally stable CN scheme, overcoming the stability limits of the conventional FDTD method and decreasing the numerical reflection of evanescent waves. The effectiveness of the proposed scheme is validated by conducting an analysis of the absorbing boundary reflection error and the simulation speed. For an ultralow frequency and small-scale problem with an extremely fine spatial mesh size ( $2\times 10^{7}$ times smaller than the minimum exited wavelength) and using time steps $4\times 10^{6}$ times larger than those in conventional FDTD, we achieved a CPU time improvement of up to $2.9360\times 10^{4}$ times compared to the conventional FDTD method, with less than 3% numerical error.