Equations of State (EOS) is crucial in simulating multiphase flows by the pseudo-potential lattice Boltzmann method (LBM). In the present study, the Peng and Robinson (P–R) and Carnahan and Starling (C–S) EOS in the pseudo-potential LBM with Exact Difference Method (EDM) scheme for two-phase flows have been compared. Both of P–R and C–S EOS have been used to study the two-phase separation, surface tension, the maximum two-phase density ratio and spurious currents. The study shows that both of P–R and C–S EOS agree with the analytical solutions although P–R EOS may perform better. The prediction of liquid phase by P–R EOS is more accurate than that of air phase and the contrary is true for C–S EOS. Predictions by both of EOS conform with the Laplace’s law. Besides, adjustment of surface tension is achieved by adjusting [Formula: see text]. The P–R EOS can achieve larger maximum density ratio than C–S EOS under the same [Formula: see text]. Besides, no matter the C–S EOS or the P–R EOS, if [Formula: see text] tends to 0.5, the computation is prone to numerical instability. The maximum spurious current for P–R is larger than that of C–S. The multiple-relaxation-time LBM still can improve obviously the numerical stability and can achieve larger maximum density ratio.

In this paper, an implementation of the complex-frequency-shifted perfectly matched layer (CPML) is developed for three-dimensional hybrid implicit-explicit (HIE) finite-difference time-domain (FDTD) method based on auxiliary differential equation (ADE). Because of the use of the ADE technique, this method becomes more straightforward and easier to implement. The formulations for the HIE-FDTD CPML are proposed. Numerical examples are given to verify the validity of the presented method. Results show that, both HIE-CPML and FDTD-CPML have almost the same reflection error, while their reflection error is about 30 dB, which is less than HIE Mur’s first-order results. The contour plots indicate that the maximum relative reflection as low as-72 dB is achieved by selecting and .

-The grounding system plays an important part in the lightning protection system of power and communication systems. The finite-difference time-domain (FDTD) method is widely used in modeling complex electromagnetic interaction problems. However, it is difficult to model the earthing conductor using the standard FDTD method in the transient grounding resistance analysis, for the electrically small depth of the earthing conductors. In this work, nonuniform grid in the FDTD methods, which is typically used to resolve fine structures, is introduced into reduce the computational domain and therefore lead to a reduction of the computational cost. To further reduce the computational memory, the uniform grids are used in the electrode length direction while non-uniform grids are occupied in the electrode sectional directions. The efficiency of the proposed model has been approved by verifying both the electromagnetic field components near the earthing conductor and the grounding resistance.Index Terms -Non-Uniform, FDTD Modeling, Earthing Conductor. I . IntroductionSensitive electronic components have been increasingly used lately both in power and communication systems. These components may suffer logic upset or damage at lower levels of induced electromagnetic interferences brought about by the lightning. As a result, evaluation of the transient grounding resistance of the grounding systems in the lightning protection systems has recently attracted considerable attentions [1]- [5].The finite-difference time-domain (FDTD) method [6], which provides a simple and efficient way of solving Maxwell' equations for a variety of problems, has been widely applied in solving many types of electromagnetic problems. It is good at predicting the electromagnetic characteristics of a particular structure for it provides extensive time-domain information and the frequency-domain information can be provided via a discrete Fourier transform.However, when the grounding resistance analysis of the lightning protection systems is involved, it will results in a huge memory and time usage with the uniform FDTD grids. Because the dimension of the grounding electrode and the reference electrode are so small compared with the total computational domain, and one has to refine the FDTD grid to the conductor dimension, which will result in huge memory usage.Several papers have introduced different methods to reduce the truncation error at grid boundary. In [7], two methods were introduced to maintain second-order accuracy. One method uses an appropriate mesh ratio between two regions to obtain the central finite differences. The other method uses a universal grading scheme with continuously variable lattice size; but a demonstration of the effectiveness of this method was not reported. The non-uniform grid technology in the FDTD method, which is typically used to resolve fine structures, can reduce the computational domain and therefore lead to a reduction of the computational cost [8]. Considering electrically small size of the grounding...

In this paper, the stability analysis of the unconditionally stable one-step leapfrog alternating direction implicit (ADI) finite-difference time-domain (FDTD) method for periodic structures is presented. The amplification matrix of the proposed leap-frog ADI-FDTD method is obtained through the spatial domain with Fourier method and eigenvalues of the Fourier amplification matrix are obtained analytically to prove the unconditional stability of the periodic leapfrog ADI-FDTD method. Numerical verification is proposed to confirm the theoretical result.