Stability analyses of non-uniform time-step and grid-size schemes for finite-difference time-domain method

The finite-difference time-domain (FDTD) method and its developments such as the Crank-Nicolson direct-splitting (CNDS), are very general numerical schemes suitable for efficient solution of a wide class of electromagnetic problems. Adopting the non-uniform time step to simulate the propagation of broadband electromagnetic waves and using the non-uniform grid size to handle with fine structures based on FDTD method can reduce the computational time and domain, further, which lead to a reduction of computational cost. However, the using of non-uniform time step (NUTS) may cause the unstable numerical results and the scheme of non-uniform grid size (NUGS) increases the truncation error at the boundary between different grid size. To remove this problem, a novel method, named to Backward-Euler direct-splitting (BEDS-FDTD), is presented. The eigenvalues of the Fourier amplification matrix method are obtained to prove the unconditional stability for BEDS-FDTD method. Meanwhile, the Complementary Derivatives Method (CDM) is introduced to address the truncation error from the variation of different grid size, further, the CDM can maintain second-order accuracy throughout the computational domain.