Implementation and Accuracy Analysis of ETD Method in FDTD Simulation Environment Bojana Nikolić, Bojan Dimitrijević, Slavoljub Aleksić University of Niš, Faculty of Electronic Engineering Niš, Serbia Mićo Gaćanović University of Banja Luka, Faculty of Electrical Engineering Banja Luka, Bosnia and Hercegovina Abstract—In this paper, the influence of two different time approximation techniques on the accuracy of the finite difference time domain (FDTD) method has been analysed. The following approximation techniques have been considered: time-average (TA) and exponential time differencing (ETD). Both algorithms have been implemented in an own developed FDTD simulation environment. In particular, an example of an on-body antenna on the human tissue is considered, as a possible case of interest. Keywords—exponential time differencing; finite difference time domain method; time-average approximation I. INTRODUCTION A rapid development of computer technology today has made the finite difference time domain (FDTD) method become the primary available tool for the design of antenna and microwave circuit components, EMC/EMI analysis, and the prediction of radio propagation [1], [2]. FDTD is a full wave time domain differential equation based technique. It is a versatile method that was proposed by Yee [3] originally for two dimensional problems with metal boundaries. Initially the FDTD method was applied to scattering problems and subsequently has become one of the most popular methods used to simulate and analyse problems in electromagnetics, ranging from antennas, microwave wave circuits, electromagnetic compatibility (EMC) issues, bioelectro- magnetics, electromagnetic scattering to novel materials and nanophotonics [1], [4], [5]. Since it is a time-domain method, solutions can cover a wide frequency range with a single simulation run. Basic advantage of this method is its simplicity and guaranteed convergence with proper choice of parameters. The disadvantages of the FDTD are high memory and computational requirements. However, this drawback becomes less and less significant, since the computational power will continue to grow exponentially in the future. Mathematical theorems for the FDTD formulation, concerning issues such as accuracy, convergence, dispersion, computational complexity and stability are available in [6]. The perfectly matched layer (PML) absorbing media [7] has proven to be the most robust and efficient technique for the termination of FDTD computational domain [8]. Its implementation, referred to as the convolutional PML (CPML), proofs to be superior over the other implementations of the PML, offering a number of advantages [9]. CPML is based on the stretched coordinate form [10] and the use of complex frequency shift (CFS) of PML parameters [11] and its application is completely independent of the host medium. In the available literature, effects of lossy media and conductors have been considered by introducing the conductivity into the original formulation of lossless FDTD update equations [12]. The temporal and spatial discretization is usually done employing second-order accurate two-point central difference method [13]. A time approximation of the components that aren’t available in required moments of time, can be performed using several different methods. One common scheme is the time-average (TA) [1], [13]. This approximation is based on the average in time between the field values at the next and the previous half time step relative to the current position in time, so one step apart. The method is second order accurate and conditionally stabile. Other variants of time approximation include the time- forward (TF) [14], time-backward (TB) [15] and the exponential time differencing (ETD) [12], [16], [17]. In the time-forward approximation a field component at one time step is approximated using its value at the previous step backward in time. In [14] this FDTD scheme is applied to the propagation analysis through a highly conductive nonlinear magnetic material. By following the pattern similar to the one of TF method, in the time-backward approximation a field component at one time step is approximated using its value at the next step forward in time. In [15] authors apply the method to the solution of the electromagnetic fields within an arbitrary dielectric scatterer of the order of one wavelength in diameter. TF and TB methods are both first order accurate. For this reason they won’t be part of the accuracy analysis presented in this paper, although both algorithms have been implemented in an own developed FDTD simulation environment. The original ETD algorithm has been applied to the simulation of electric conductive media and isotropic lossy dielectrics [16], [18]. Subsequently, the ETD schemes for the simulation of wave propagation in magnetized plasma are developed [19]. In [19], the original approximations of ETD by Taylor series schemes are presented. The ETD method has a second-order temporal accuracy. In [16] authors determine the stability condition and analyse the accuracy of the exponential and average time- approximation schemes for FDTD in an isotropic, homogeneous lossy dielectric with electric and magnetic