The numerical superiority of the unconditionally stable fundamental locally one-dimensional finite-difference time-domain (FLOD-FDTD) method has been proved in some studies. In this paper, the FLOD-FDTD method is extended to simulate general dispersive media in combination with the auxiliary differential equation (ADE) scheme with formulation based on the electric field ( E ) and electric polarization ( P ). The relative permittivity of dispersive media is described by the arbitrary multi-term modified Lorentz model which can also represents Debye and Drude models. Thus, this method is able to include different types of dispersion in a unified formulation. The correctness and efficiency of the proposed method is demonstrated by some numerical cases.

The preceding works introduced the leapfrog complying divergence implicit finite-difference time-domain (CDI-FDTD) method, which exhibits high accuracy and unconditional stability. In this study, the method is reformulated to simulate general electrically anisotropic and dispersive media. The auxiliary differential equation (ADE) method is employed to solve the equivalent polarization currents, which are then integrated into the CDI-FDTD method. The iterative formulae are presented, and the calculation method is similar to that of the traditional CDI-FDTD method. Additionally, the Von Neumann method is utilized to analyze the unconditional stability of the proposed method. To evaluate the performance of the proposed method, three numerical cases are conducted. These include calculating the transmission and reflection coefficients of a monolayer graphene sheet and a monolayer magnetized plasma, as well as the scattering properties of a cubic block plasma. The numerical results obtained by the proposed method demonstrate its accuracy and efficiency in simulating general anisotropic dispersive media, compared to both the analytical method and the traditional FDTD method.

The recently proposed complying-divergence implicit finite-difference time-domain (CDI-FDTD) method was demonstrated with relatively superior numerical performance, especially with the unconditional stability of implicit methods and the complying-divergence property. In addition, the complying-divergence property was also extended to and successful utilized in semi-implicit FDTD methods for significantly more efficient and accurate electromagnetic (EM) computations with models containing fine structures in one or two directions. This work offers an effective implementation of the complex frequency-shifted perfectly matched layer (CFS-PML) based on a one-step leapfrog CD-hybrid implicit-explicit FDTD (CD-HIE-FDTD) method with more robust simulations to address more complicated open-domain EM challenges. The stretching factors are initially introduced into Maxwell’s equations to derive a compact first-order differential matrix form. Subsequently, the matrix is subjected to space-time operator splitting to establish a complying-divergence framework with a two-step iterative solution. For more numerical efficiency, we utilize the leapfrog time-stepping strategy to eliminate intermediate variables and get a one-step iterative solution. Moreover, for the aforementioned computational framework, we offer an analysis for the terms of numerical stability and complying-divergence property. Meanwhile, the various examples given in this work also verify the correctness and efficacy of the proposed method.

This study introduces a novel, unified arbitrary-order matrix exponential-symplectic FDTD^(p,q) (ME-SFDTD^(p,q)) method tailored for simulating anisotropic time-varying plasma medium. The governing equations, comprising both current density and Maxwell’s equations, are reformulated into a compact first-order differential matrix form and then discretized in the temporal domain using a multistage p-order symplectic integrator. Meanwhile, the matrix exponential (ME) coefficients arising from the symplectic discretization are efficiently computed utilizing the ME method, while spatial derivatives are approximated by a q-order central-difference scheme. These processes yield a complete numerical iteration framework specific to the ME-SFDTD^(p,q) format, optimized for field components calculations in an anisotropic time-varying plasma medium. To effectively truncate the simulation space modeled by the ME-SFDTD^(p,q) method, a high-performance complex frequency-shifted perfectly matched layer (CFS-PML) technique is proposed, which shares the same primary iteration formulas as the ME-SFDTD^(p,q) method, exhibiting excellent absorption performance and high compatibility. In addition, a comprehensive numerical stability and dispersion analysis is conducted to confirm the feasibility of the proposed method. Finally, several numerical examples are presented to thoroughly validate the accuracy and efficiency of the approach.

This study presents an implicit unconditionally stable and divergence-preserving alternating-direction-implicit finite-difference time-domain (DP-ADI-FDTD) method for modeling the multi-pole lossy Debye dispersive medium, developed using the Z-transform technique. To address ground-penetrating radar (GPR) problems with enhanced computational efficiency and accuracy, the DP-ADI-FDTD method is integrated with the traditional FDTD method, culminating in an advanced hybrid subgrid framework. In this approach, electromagnetic field components in the coarse grid region are computed using the explicit FDTD scheme, while those in the fine grid region are solved using the implicit DP-ADI-FDTD scheme. The unconditional stability of the DP-ADI-FDTD method permits the use of a uniform time step on both coarse and fine grids, thereby improving computational efficiency and simplifying numerical implementation. The proposed Z-transform-based DP-ADI-FDTD method is validated through theoretical analysis and numerical experiments, demonstrating its divergence-preserving property, unconditional stability, and the effectiveness of the extended hybrid subgrid technique.

Ground penetrating radar (GPR) forward modeling is one of the core geophysical research topics and also the primary task of simulating ground penetrating radar system. It is a process of simulating the propagation laws and characteristics of electromagnetic waves in simulated space when the distribution of internal parameters in the exploration region is known. And the finite-difference-time-domain (FDTD) method has the characteristics of simulating the space-time transient evolution of electromagnetic wave, whose numerical method is simple and easy to program, so it has become one of the most extensively utilized methods in GPR forward modeling. It is generally known that the conventional FDTD approach requires finer uniform Yee cell all the time to produce satisfactory accuracies from numerical simulations of the GPR. However, the smaller temporal incremental has to be adopted due to the lower spatial incremental, which would dramatically weaken the advantage of the FDTD method. To solve this issue, the subgridding-technique-based hybrid local-one-dimensional FDTD (LOD-FDTD) is applied in this work to modeling the classical GPR scenarios. In this method, the unconditional-stable LOD-FDTD is employed in the fine-grid domain, while the traditional FDTD is used in the coarse-grid domain, which could avoid the oversampling problem in the local domain if the uniform fine-grid scheme is adopted. Meanwhile due to the unconditional stability of the LOD-FDTD, the larger time step, derived from the coarse grid which satisfies the Courant-Friedrichs-Lewy (CFL) stability condition, could be utilized in the whole domain so that the long-time interpolation process could be circumvented. Additionally, the proposed approach could be arbitrarily adjusted by means of different ratio of both coarse- and fine-grid, and hence it holds much higher generality. As compared with the auxiliary differential equation (ADE) technique, the Z-transform method is integrated into FDTD methods for modeling multi-pole Debye-based dispersive media in this method, resulting in more direct numerical implementations and fewer computing steps. Finally, three different classical GPR problems are carried out to validate accuracies and efficiencies of the proposed method.