An Adaptive Time Step FDTD Method for Maxwell’s Equations

This letter is concerned with a new finite difference method of the 2-D Maxwell's equations in time domain by using adaptive time steps (called ATS-FDTD). First, based on the Yee's staggered points and the central difference formulas for spatial derivatives, the Maxwell's equations are reduced into a system of ordinary differential equations (ODEs). Second, the continuous field functions of time in the system are approximated by Taylor's polynomials of high and adaptive degrees. Then, an algorithm for computing coefficients of the polynomials is proposed and, finally, the ATS-FDTD method is formed. It is shown that ATS-FDTD is second order accurate in space and any order accurate in time. By analyzing the stability domain of the system of ODEs, a criterion to select adaptively the discretizing time steps and the accuracy in time (or the degree of polynomials) is provided and the stability of ATS-FDTD is proven. Numerical experiments to compute the errors, test energy conservation, and simulate a wave propagation generated by a point source in a waveguide are carried out. Computational results confirm validity of the method.

I. Introduction

It is well known that the finite-difference time-domain (FDTD) method (also called the Yee Scheme) [1] is a very popular and efficient method for numerical solutions of Maxwell's equations and applied to a broad ranges of problems in electromagnetics [2]. In FDTD the way of time marching is frog-jumping and the time step is confined to the Courant-Friedrichs-Lewy (CFL) condition [3]. To overcome this limitation, the alternating direction implicit FDTD (ADI-FDTD) methods [4], [5] were proposed by implicitly discretizing the equations and proven that the time step does not depend on the spatial increments [4]–[6]. However the time step size is limited by the errors from numerical dispersion and the accuracy of time [7]. Improvement of the time accuracy of ADI-FDTD by extrapolation can be seen in [8]. Thus, it is important to study time marching ways and methods to improve the accuracy in time.