Leapfrog Alternating-direction-implicit Finite-difference Time-domain Method Research Articles

Unified Formulation of Leapfrog Complying Divergence Implicit FDTD Method for Lossy Media

The leapfrog alternating direction implicit finite-difference time-domain (ADI-FDTD) method for lossy media has been found to suffer from the non-physical field leakage problem in the non-penetrable structures, which gives rise to inaccuracies in modelling lossy media during scattering analyses. To address such problem, in this paper, we propose a unified formulation of leapfrog complying-divergence implicit (CDI)-FDTD method for lossy media without field leakage problem. Several different lossy schemes to incorporate the conductivity terms of lossy media, including exponential time differencing, averaging and forward-forward schemes are integrated into a unified formulation. The proposed leapfrog CDI-FDTD method not only resolves the problem of field leakage but also has complying divergence. Moreover, since the implicit update procedures in the proposed formulation are in the fundamental form with matrix-operator-free right-hand sides, the floating-point operations (flops) are fewer than that of the conventional leapfrog ADI-FDTD formulation for lossy media. Numerical results are given to demonstrate the advantages of the proposed method over the leapfrog ADI-FDTD method in accuracy, efficiency, complying divergence and resolving the field leakage problem for lossy media.

From Time-Collocated to Leapfrog Fundamental Schemes for ADI and CDI FDTD Methods

The leapfrog schemes have been developed for unconditionally stable alternating-direction implicit (ADI) finite-difference time-domain (FDTD) method, and recently the complying-divergence implicit (CDI) FDTD method. In this paper, the formulations from time-collocated to leapfrog fundamental schemes are presented for ADI and CDI FDTD methods. For the ADI FDTD method, the time-collocated fundamental schemes are implemented using implicit E-E and E-H update procedures, which comprise simple and concise right-hand sides (RHS) in their update equations. From the fundamental implicit E-H scheme, the leapfrog ADI FDTD method is formulated in conventional form, whose RHS are simplified into the leapfrog fundamental scheme with reduced operations and improved efficiency. For the CDI FDTD method, the time-collocated fundamental scheme is presented based on locally one-dimensional (LOD) FDTD method with complying divergence. The formulations from time-collocated to leapfrog schemes are provided, which result in the leapfrog fundamental scheme for CDI FDTD method. Based on their fundamental forms, further insights are given into the relations of leapfrog fundamental schemes for ADI and CDI FDTD methods. The time-collocated fundamental schemes require considerably fewer operations than all conventional ADI, LOD and leapfrog ADI FDTD methods, while the leapfrog fundamental schemes for ADI and CDI FDTD methods constitute the most efficient implicit FDTD schemes to date.

Improved Hybrid Leapfrog ADI-FDTD Method for Simulating Near-Field Coupling Effects Among Multiple Thin Wire Monopole Antennas on a Complex Platform

A hybrid one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method integrated with a thin wire algorithm is developed for simulating the near-field mutual coupling effects among multiple wire antennas on an electrically large warship platform, where induced surface current and electric field distributions are also captured. Its stability analysis, based on von Neumann method, is performed in a semi-analytical way and verified numerically. In order to demonstrate its accuracy and efficiency, the near-field mutual coupling effect between two thin wire antennas is at first investigated. Then, it is applied to accurately characterize complex electromagnetic environment effects on a warship with multiple thin wire antennas, and the results are compared with those obtained by the other commercial simulators.

An Efficient Domain Decomposition Parallel Scheme for Leapfrog ADI-FDTD Method

A flexible and universal domain decomposition parallel scheme is proposed for the unconditionally stable finite-difference time-domain (FDTD) method. The leapfrog alternating direction implicit FDTD (ADI-FDTD) method is employed to eliminate the restriction of the Courant–Friedrichs–Lewy stability condition. The proposed domain decomposition parallel implementation of the leapfrog ADI-FDTD method is more flexible with process allocation and requires fewer data communications. A buffer region is introduced to decouple the interactions between neighboring subdomains at each time step. Electromagnetic simulations are presented to demonstrate the applicability, accuracy, and efficiency of the proposed method.

Scattered Field Leapfrog ADI–FDTD Method for Drude Dispersive Media

This letter presents a new unconditionally stable scattered field (SF) finite-difference time-domain (FDTD) scheme for Drude dispersive medium based on leapfrog alternating direction implicit (ADI) technique. The proposed scheme introduces polarization current density and analytical incident field terms in the leapfrog scheme without adding extra computational complexities except causing minor changes in the coefficients for the central spatial locations. Unconditional stability and accuracy of the scheme are validated by numerical tests. When the Courant number is set to three or a higher value, the scheme can perform faster than the explicit FDTD scheme.

Broadband modeling surface plasmon polaritons in optically pumped and curved graphene structures with an improved leapfrog ADI-FDTD method

An improved one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is proposed to study surface plasmon polaritons (SPPs) in optically pumped and electrostatic applied curved graphene structures, with their intraband and interband surface conductivities modeled with the vector fitting technique. Such a method can quickly capture the broadband characteristics of SPPs propagating along monolayer, bilayer and trilayer graphene sheets of arbitrary shapes. Numerical stability and accuracy are validated through characterizing field distributions of the SPPs supported in the 180°-curved and spiral graphene sheet waveguides and higher computational efficiency are achieved in comparison with the conventional FDTD method.

Efficient unconditionally stable one‐step leapfrog ADI‐FDTD method with low numerical dispersion

An efficient unconditionally stable one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method based on the controlling parameters is presented. First, three controlling parameters are introduced to the matrices of the Maxwell's equations to decrease the numerical dispersion error, and then the formulation of an efficient one-step leapfrog ADI-FDTD method is derived. Second, the analysis shows that the proposed method is unconditionally stable. Moreover, the numerical dispersion relation of the proposed method is derived analytically. Third, the process of determination of the controlling parameters is shown. Furthermore, the effects of the propagation angle, mesh size, time step and frequency on the dispersion characteristics of the proposed method are also investigated. The result shows that the normalised numerical phase velocity error (NNPVE) and maximum NNPVE of the proposed method are decreased significantly. Finally, two numerical examples are simulated to demonstrate the accuracy and efficiency of the proposed method.

One-Step Leapfrog ADI-FDTD Method in 3-D Cylindrical Grids With a CPML Implementation

A three-dimensional (3-D) unconditionally stable one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method in the cylindrical coordinate system is presented. It is more computationally efficient while preserving the properties of the two-step scheme. By reusing the auxiliary variable, it also uses less memory than the two-step scheme. In contrast with the one-step leapfrog ADI-FDTD method in the Cartesian coordinate system, some implicit equations of the one-step leapfrog ADI-FDTD method in the cylindrical coordinate system are not tridiagonal equations. The Sherman Morrison formula is used to solve them efficiently. To solve open region problems, convolutional perfectly matched layer (CPML) for the one-step leapfrog scheme is proposed. Numerical results are presented to validate them.

High-Order Unconditionally Stable Two-Step Leapfrog ADI-FDTD Methods and Numerical Analysis

High-order unconditionally stable two-step leapfrog alternating direction implicit-finite-difference time-domain (ADI-FDTD) methods in three-dimensional (3-D) domains are presented. Based on the exponential evolution operator (EEO), the Maxwell's equations in a matrix form can be split into four subprocedures first, and then two subprocedures are generated by using the leapfrog scheme. Subsequently, the proposed high-order methods are theoretically proven for unconditional stability, and the numerical dispersion is derived analytically. There are several factors to effect the dispersion: for example, the order of schemes, propagation angle, time step, and mesh size. These four factors are illustrated comprehensively in this paper. Specifically, the normalized numerical phase velocity error (NNPVE) of the proposed schemes is lower than that of the one-step leapfrog ADI-FDTD method. Finally, numerical experiments are presented to demonstrate the validity of the proposed methods. It was found that the proposed second-order method has the same level of relative error as that of the four-step ADI-FDTD method, but it has a higher computational efficiency.

One-step leapfrog ADI-FDTD method for simulating electromagnetic wave propagation in general dispersive media

The one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is reformulated for simulating general electrically dispersive media. It models material dispersive properties with equivalent polarization currents. These currents are then solved with the auxiliary differential equation (ADE) and then incorporated into the one-step leapfrog ADI-FDTD method. The final equations are presented in the form similar to that of the conventional FDTD method but with second-order perturbation. The adapted method is then applied to characterize (a) electromagnetic wave propagation in a rectangular waveguide loaded with a magnetized plasma slab, (b) transmission coefficient of a plane wave normally incident on a monolayer graphene sheet biased by a magnetostatic field, and (c) surface plasmon polaritons (SPPs) propagation along a monolayer graphene sheet biased by an electrostatic field. The numerical results verify the stability, accuracy and computational efficiency of the proposed one-step leapfrog ADI-FDTD algorithm in comparison with analytical results and the results obtained with the other methods.

Analysis of the Divergence Properties for the Three-Dimensional Leapfrog ADI-FDTD Method

Analysis of the divergence properties for the three-dimensional (3-D) leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is presented in this paper. The analysis of divergence properties is based on the Fourier domain plane wave method for dispersion analysis but extended with additional procedure to determine the homogeneous solutions. The singular value decomposition (SVD) is used to obtain the solutions of the homogeneous system in a reliable manner. There are two linearly independent null-modes in the homogeneous system corresponding to the repeated zero singular values. To provide more physical insights, proper linear combinations of the two null-modes are determined as TE and TM modes with respect to the direction of discrete numerical space wavevector. The divergence of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E</i> fields is analyzed by examining the TM mode, while the divergence of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> fields is analyzed by examining the TE mode. Numerical results for the divergence properties of the leapfrog ADI-FDTD method are presented and compared with the conventional ADI-FDTD method, along with numerical experiments to illustrate the existence of divergent waves. The leapfrog ADI-FDTD method is not divergence-free in the source-free region, as is the case for the conventional ADI-FDTD method. Furthermore, both methods have different divergence properties in general.

An Unconditionally Stable One-Step Arbitrary-Order Leapfrog ADI-FDTD Method and Its Numerical Properties

An one-step arbitrary-order leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method is presented. Its unconditional stability is analytically proven and numerically verified. Its numerical properties are shown to be identical to those of the conventional ADI-FDTD and LOD-FDTD methods. However, its computational expenditure is found to be the lowest. In other words, in comparison with the ADI-FDTD and LOD-FDTD methods, the one-step arbitrary-order leap-frog ADI-FDTD method retains identical numerical modeling accuracy but with higher computational efficiency.

One-Step Leapfrog ADI-FDTD Method Including Lumped Elements and Its Stability Analysis

A one-step leapfrog alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method including lumped elements is presented. The method is originated from the conventional ADI-FDTD method, but without mid-time-step computations. Its unconditional stability is analytically proven by combining the von Neumann method with the Jury criterion. In addition, its unconditional stability and high computational efficiency are verified through numerical experiments.