Unconditionally stable finite-difference time-domain methods and their applications

The finite-difference time-domain (FDTD) method and its variations have been widely used in simulating electromagnetic structures that include RF/microwave/optical circuits. However, due to the use of explicit march-in-time schemes, their time steps are limited not only by the modeling accuracy but also by the Courant-Friedrich-Levy (CFL) stability condition. This CFL condition poses quite a significant restriction on the application of the FDTD methods to a number of problems, in particular those requiring highly non-uniform grids. In this talk, we present an overview of the two unconditionally stable FDTD schemes that have been developed in recent years. The first one is the alternating-direction-implicit (ADI) FDTD method and the second one is the FDTD method using Laguerre polynomials. We show the principles of these two methods and analyze the associated errors and the techniques to improve them. Finally, we demonstrate the advantages of the two methods by showing their applications such as computation of conducting loss and a PEC slot.

I. Introduction

Maxwell's equations, which represent a fundamental relationship between electric and magnetic fields, have been studied for decades. Although the analytical solutions have been given for many structures, it is not easy to obtain the analytical expressions for predicting the field behavior in the complex structures and composite materials. The alternative way is to use numerical techniques which have been under study and proven to be effective and efficient for solving Maxwell's equations in both time domain and frequency domain. With the particular desire of obtaining the full wave analysis, research has been driven into finding novel time domain techniques. The Finite-difference Time-domain (FDTD) method is one of the most popular time domain methods. It has been extensively studied and employed due to its simplicity, effectiveness and flexibility [1]. For the electrically large structures and highly conductive materials, however, the conventional FDTD algorithm requires large computation resources and prohibitively long simulation time owing to its two inherent limits: dispersion errors and the Courant-Friedrich-Levy (CFL) stability condition. Consequently, the recent FDTD-based algorithm developments have been aimed at removing or alleviating the two constraints. They include multiresolution time-domain (MRTD) method with low dispersion [2], pseudospectral time-domain (PSTD) method with great computation savings [3], and the recently developed unconditionally stable FDTD algorithms with the complete removal of CFL condition. In this paper, we will focus on the unconditionally stable algorithms.