Low numerical dispersion error ADI-FDTD methods

Abstract

The Finite Difference Time Domain (FDTD) method is one of the most popular time domain methods in computational electromagnetics. The FDTD method is easy to implement and a wide band solution can compute from a single run of simulation. However, the application of the FDTD method is limited by the requirement of the computational resources. Such requirement is due to the numerical dispersion error and the Courant-Friedrich-Levy (CFL) stability condition, which relates the choosing of the cell size and the time-step. In recent research, an unconditionally stable FDTD method was proposed - one that can set the time-step to any arbitrary value without compromising the stability of the system. This method applies the Alternating Direction Implicit (ADI) technique to solve the finite difference equations; therefore, it is named the 'ADI-FDTD method." The ADI-FDTD method is useful to simulate a structure with fine features because the time-step can be set to the desired value, based on the signal but not the smallest cell size. However, it suffers a drawback in that the numerical dispersion error is found to increase when the ADI technique is applied. In this thesis, two modifed ADI-FDTD methods are proposed to reduce the numerical dispersion error. The first method is the high-order ADI-FDTD method, which employs the multi-points high-order central difference scheme to approximate the spatial derivative terms. This method is still unconditionally stable and can reduce the numerical dispersion. However, it is found that the numerical dispersion error of the sixth-order ADI-FDTD method is close to the limit of the conventional ADI-FDTD method, and the improvement is found to be relatively insignificant when the time-step is large. This motivated the development of the second method called (2,4) low numerical dispersion(LD) ADI-FDTD method. This method is based on the fourth-order ADI-FDTD method. The coefficients of finite difference operator are modified by minimizing the error terms in the numerical dispersion relation. This modification does not affect the unconditionally stable property. In addition, the (2,4) LD ADI-FDTD method can provide a significant wide band reduction on the numerical dispersion error for any time-step. Furthermore, there is an alternative scheme that can reduce the numerical dispersion error at a specified propagation angle.