Piecewise Calculation Scheme for the Unconditionally Stable Chebyshev Finite-Difference Time-Domain Method

The unconditionally stable (US) Chebyshev (CS) finite-difference time-domain (FDTD) method is extended for solving the problems of long-time simulation or harmonic resonance using a piecewise calculation scheme. First, the CS differential matrix is derived from the inversion of integral matrix with the differential characteristic of the CS polynomials. Then, the 2-D CS FDTD formula with initial values is derived based on the CS differential matrix, highlighting the merits of a closed interval for the CS basis functions and the 0th-order CS polynomial. In addition, the time-frequency support of CS functions for order selection is discussed. Finally, based on the above derivation, a piecewise calculation scheme is proposed to simulate an entire time, where the electromagnetic field is reconstructed piecewise with the Clenshaw law. Numerical examples for the 2-D ${\mathrm { TE}}_{z}$ case show that the proposed method agrees well with the conventional FDTD method with the relative difference lower than −50 dB. This represents a higher accuracy than the associated hermite (AH) FDTD method and reduces the memory compared with the original CS FDTD method.