Conductivity introduction to 3D Locally One-Dimensional FDTD Method

The Finite Difference Time Domain (FDTD) method [1] has been commonly used for the numerical simulation of electromagnetic waves in the time domain. The FDTD method is robust and flexible and its accuracy is well relevant with the fine spatial sampling (∆s). However, the time step size (∆t) is limited by the Courant-Friedrich-Levy (CFL) stability condition [1]. When ∆s has to be small for the modelling, the maximum temporal sampling under the CFL stability condition (∆tCFL) becomes unreasonably small. Thus the explicit scheme is confronted by the computational inefficiency. The Alternating Direction Implicit (ADI) FDTD method [2] [3] [4] has been introduced for overcoming CFL stability condition, but the ADI-FDTD method requires more CPU time than the explicit FDTD method per FDTD iteration [5]. Therefore, the ADI-FDTD method demands high CFL Number (CFLN), defined as ∆t / ∆tCFL, to improve the computational efficiency. A Locally One Dimensional (LOD) FDTD method [6] [7] [8] [9] is an alternative implicit method, aiming at the reduction of the CPU time per FDTD of the ADI-FDTD method so that the high computational efficiency can be achieved by the relatively low CFLN . Although the number of equations to be computed with the LOD-FDTD method is the same as that for the conventional ADI-FDTD method, the arithmetic operations are fewer. Therefore it is more efficient overall than the ADI-FDTD method in computational time [10]. The unconditional stability of the 3D LOD-FDTD method has been proven theoretically and validated numerically [11]. Also the accuracy of the LOD-FDTD method can be improved by adopting a different splitting schemes such as the strange method. Although the extra step may cost additional time, the LOD-FDTD method with a different splitting scheme can yield a better relative accuracy than the ADI-FDTD method for the same ∆t [12]. The LOD-FDTD method has been mostly considered in a lossless medium case. However, lossy media which has Direct current (DC) conductivity (σ) parameter is essential for real applications. In this paper, conductivity term is introduced into two different types of 3D LOD-FDTD methods and both methods are numerically compared. The 3D LOD-FDTD method with different basic Maxwell’s equations significantly affects the experiential results especially when the conductivity is greater than zero and the CFLN is more than one. Section II proposes two approaches to include conductivity term into the 3D 3-steps LOD-FDTD method. Section III performs the numerical experiments to quantify these accuracy.