Precise Integration Time Domain Method Research Articles

A Switched-Huygens-Subgridding-Based Combined FDTD–PITD Method for Fine Structures

A new combined method is proposed for the problems containing fine structures. In this method, the precise-integration time-domain (PITD) method is adopted for fine structures with fine cells, and the finite-difference time-domain (FDTD) method is used for the rest with coarse cells. The fields in the coarse and fine cells are updated synchronously, which makes the proposed method fast. Besides, since the PITD method is only applied locally, the limitation of its large memory requirement is also alleviated. The numerical results confirm that this combined method has a fast computational speed.

An Efficient Numerical Formulation for Wave Propagation in Magnetized Plasma Using PITD Method

A modified precise-integration time-domain (PITD) formulation is presented to model the wave propagation in magnetized plasma based on the auxiliary differential equation (ADE). The most prominent advantage of this algorithm is using a time-step size which is larger than the maximum value of the Courant–Friedrich–Levy (CFL) condition to achieve the simulation with a satisfying accuracy. In this formulation, Maxwell’s equations in magnetized plasma are obtained by using the auxiliary variables and equations. Then, the spatial derivative is approximated by the second-order finite-difference method only, and the precise integration (PI) scheme is used to solve the resulting ordinary differential equations (ODEs). The numerical stability and dispersion error of this modified method are discussed in detail in magnetized plasma. The stability analysis validates that the simulated time-step size of this method can be chosen much larger than that of the CFL condition in the finite-difference time-domain (FDTD) simulations. According to the numerical dispersion analysis, the range of the relative error in this method is 10−6 to 5×10−4 when the electromagnetic wave frequency is from 1 GHz to 100 GHz. More particularly, it should be emphasized that the numerical dispersion error is almost invariant under different time-step sizes which is similar to the conventional PITD method in the free space. This means that with the increase of the time-step size, the presented method still has a lower computational error in the simulations. Numerical experiments verify that the presented method is reliable and efficient for the magnetized plasma problems. Compared with the formulations based on the FDTD method, e.g., the ADE-FDTD method and the JE convolution FDTD (JEC-FDTD) method, the modified algorithm in this paper can employ a larger time step and has simpler iterative formulas so as to reduce the execution time. Moreover, it is found that the presented method is more accurate than the methods based on the FDTD scheme, especially in the high frequency range, according to the results of the magnetized plasma slab. In conclusion, the presented method is efficient and accurate for simulating the wave propagation in magnetized plasma.

Numerical Treatment for Electromagnetic Wave in Time-Variant Medium Using Generalized PITD Method

In this letter, the propagation of an electromagnetic (EM) wave in a time-variant medium is treated with a novel methodology stemming from the precise integration time domain (PITD) method. This methodology is a generalized version of the conventional PITD method (thus GPITD, for short). In the GPITD method, the electric displacement and the magnetic induction serve as the iteratively calculated components. In order to overcome the difficulty in implementing the precise integration (PI) routine caused by the time variation of the original coefficient matrix, an effective piecewise constant coefficient matrix is constructed. Both the numerical dispersion relation and the numerical stability condition of the GPITD method are described. Successful application of the GPITD method to solve the EM propagation problem associated with a layer whose permittivity suffers: 1) an abrupt change to a new value and 2) a sinusoidal modulation, confirms its effectiveness.

Analysis of the propagation properties of photonic crystals with defect by the precise integration time domain method

ABSTRACTPropagation properties of photonic crystals with defect are analyzed by the precise integration time domain (PITD) method in the paper. The precise integration is applied to the finite difference time domain in this method. The Yee cell differential discretization technique is used in space, and the precise integration method is adopted in time. The correctness of the method is demonstrated by calculating the one-dimensional photonic crystals with defect. The transmission coefficients of the transverse electric (TE) and transverse magnetic (TM) modes in the two-dimensional photonic crystals with defect are also obtained by the method. Furthermore, the stability, precision and efficiency of the method are discussed. Numerical results reveal that the PITD method is not limited by the Courant stability condition and has higher calculation precision and efficiency, which provides a new and effective analytical method for studying the propagation properties of photonic crystals with defect.

Stability Condition and Numerical Dispersion of Fourth-Order Compact Precise-Integration Time-Domain Method

In order to mitigate the numerical dispersion errors of the compact precise integration time-domain (CPITD) algorithm, the fourth-order CPITD [CPITD(4)] method is proposed recently which is based on the fourth-order finite-difference scheme and the precise integration technique. Both the stability condition and the numerical dispersion equation have been obtained. In this paper, the numerical dispersion characteristics of the CPITD(4) method, especially the improvement of the numerical dispersion by using the fourth-order finite-difference scheme are discussed in detail. It is found that the numerical dispersion errors of the CPITD(4) method are much less than those of the CPITD, compact finite-difference time-domain (CFDTD), and CFDTD(2,4) methods and can be made nearly independent of the time-step size. Furthermore, the numerical experiments validate the accuracy and the efficiency of the CPITD(4) method, and also verify our analysis of the numerical dispersion characteristics of the CPITD(4) method.

Conformal Precise-Integration Time-Domain Method for Analyzing Electromagnetic Fields in Fine-Structured Device With Moving 3-D Part

Herein, a novel method, referred to as conformal precise-integration time-domain (CO-PITD) method, which combines the precise-integration time-domain (PITD) method and the conformal mesh technique, is proposed to numerically calculate the evolving electromagnetic (EM) fields in the fine-structured device with moving part in time domain, particularly for 3-D case. The EM parameters in the computation cells partially occupied by moving medium are evaluated with the aid of the conformal mesh technique. For the implementation of the PITD method, the discretized Maxwell equations in those cells are modified due to the temporal variation, caused by the movement, of the EM parameters. Furthermore, the challenge from time-variant coefficient matrix is overcome by constructing an effective piecewise time-constant substitute. Meanwhile, the stability condition and the dispersive relation of the proposed method are both theoretically derived. The validity of the CO-PITD method is verified in the context of several numerical experiments. In addition, the comparisons with the established methods show that the proposed method is of excellent efficiency since a much large time step size is permitted in it.

A Memory-Saving Realization of the Perfectly Matched Layer in the Precise-Integration Time-Domain Method

The memory demand of the precise-integration time-domain (PITD) method is enlarged quite a lot by the implementation of the perfectly matched layers following the conventional way, which is enclosing the field components of the perfectly matched layer (PML) regions and the main region into one field component vector. In this letter, a region-splitting (RS) technique, for the memory-saving realization of the PML in the PITD method, is proposed. In the RS technique, the field components in the main region and each PML region are calculated separately. The accurate mutual information transmissions between any two regions are guaranteed by advancing the sampling time points of the connecting field components by half-time-step. Several numerical experiments are performed to verify the effectiveness and the efficiency of the proposed technique.

Numerical analysis of electromagnetic field in devices with the high-speed unit

PurposeThe purposes of this paper are to numerically analyse the distribution of the electromagnetic field in the electromagnetic device wherein a high-speed unit exists and to develop a strong tool to analyse the evolution of an electromagnetic field tangled with moving parts.Design/methodology/approachThe precise integration time domain (PITD) method and parameter weighted averaging approximation scheme.FindingsIt is shown that that the electromagnetic field in the device is significantly affected by the velocity of the moving unit and the parameters of the base material. The computation resources of the proposed method are saved and the efficiency is enhanced.Originality/valueThe parameter approximation (PA)-PITD method can be an effective and efficient time domain method to analyse the evolution of the electromagnetic field in electromagnetic devices with moving parts and similar problems.

Numerical Analysis of Electromagnetic Scattering From a Moving Target by the Lorentz Precise Integration Time-Domain Method

A novel numerical method, referred to as Lorentz precise integration time-domain (Lorentz-PITD) method, is proposed to deal with the scattering problem from a moving conducting slab. Both the overset grid generation technique and the Lorentz transformation are employed in this method. By using the Lorentz transformation and the linear interpolation technique, the incident plane wave in the rest frame is introduced to the moving frame; the scattered fields are transformed from the moving frame to the rest frame. Numerical experiments validate the Lorentz-PITD method and show that the Lorentz-PITD method is more computationally efficient compared with the Lorentz finite-difference time-domain method. An accompanied find is that there must be a tradeoff between the applicable frequency range and the computation cost in the proposed method.

A Low-Dispersion Realization of a Rectangular Grid With PITD Method Through Artificial Anisotropy

Through using a nonuniform rectangular grid in the time-domain algorithm, the computational efficiency can be improved obviously, but the numerical anisotropic dispersion error is seriously deteriorative. In this letter, a rectangular grid with the precise-integration time-domain method through artificial anisotropy is proposed to reduce the numerical dispersion error of a rectangular grid. Both the stability condition and the numerical dispersion equation are obtained analytically, and the numerical anisotropic dispersion of the proposed method is examined in detail. It is found that the numerical dispersion error can be reduced obviously and can also be made nearly independent of the time-step size. The numerical experiments validate and verify that the proposed method is of higher accuracy and efficiency.

A split-step-scheme-based precise integration time domain method for solving wave equation

Purpose – This paper aims to present a modified precise integration time domain (PITD) method for the numerical solution of 2D scalar wave equation. Design/methodology/approach – The split step (SS) scheme is applied to factorize the conventional PITD calculation into two sub-steps procedures and then field components can be updated along one spatial direction only in each sub-step. The perfectly matched layer (PML) absorber is extended to this method for modeling open region problems by using the stretched coordinate approach. Findings – It is shown that this method requires less computation time and storage space in comparison with the conventional PITD method, yet maintains the numerical stability despite using large time steps. Research limitations/implications – The WE-PITD method requires the divergence free region, which may be a limit on its usage. Hence, there is a challenge of using this technique in the 3D problems. Originality/value – Based on the SS scheme, the PITD method is used to solve the scale wave equation rather than Maxwell's equations, leading to a significant reduction in the computation time and memory usage.

An efficient application of PML in fourth-order precise integration time domain method for the numerical solution of Maxwell's equations

Purpose – The purpose of the paper is to introduce a perfectly matched layer (PML) absorber, based on Berenger's split field PML, to the recently proposed low-dispersion precise integration time domain method using a fourth-order accurate finite difference scheme (PITD(4)). Design/methodology/approach – The validity and effectiveness of the PITD(4) method with the inclusion of the PML is investigated through a two-dimensional (2-D) point source radiating example. Findings – Numerical results indicate that the larger time steps remain unchanged in the procedure of the PITD(4) method with the PML, and meanwhile, the PITD(4) method employing the PML is of the same absorbability as that of the finite-difference time-domain (FDTD) method with the PML. In addition, it is also demonstrated that the later time reflection error of the PITD(4) method employing the PML is much lower than that of the FDTD method with the PML. Originality/value – An efficient application of PML in fourth-order precise integration time domain method for the numerical solution of Maxwell's equations.

An Efficient 2-D Compact Precise-Integration Time-Domain Method for Longitudinally Invariant Waveguiding Structures

Based on both the compact technique and the precise-integration (PI) technique, a 2-D compact precise-integration time-domain method (CPITD) is developed in order to mitigate the rapidly growing numerical dispersion errors of a recently proposed compact finite-difference time-domain (FDTD) algorithm with increased time-step size when modeling electrically large and longitudinally invariant waveguiding structures. The stability condition and the dispersion equation of the new algorithm are both derived analytically. The provided enhancement over the FDTD, compact FDTD, and the conventional PITD methods is exhibited through theoretical examination of the dispersion performance, and subsequently, validated by means of numerical experimentation. It is found that with the PI technique, the maximum limit of the time step allowable by the new algorithm's stability criterion is much larger than the Courant-Friedrich-Levy limit of the compact-FDTD method, more particularly, numerical dispersion errors can be made nearly independent of time-step size, i.e., an appreciable reduction of numerical dispersion error is achievable at any time-step size in the simulations. Numerical experimentations of typical waveguide structures verify and validate the very promising theoretical results. This CPITD algorithm will be very useful in electrically large and longitudinally invariant waveguiding structures since the decreased number of grid points in the 2-D domain greatly reduces the memory requirements and also the overall computational time, and the PI technique nearly removes the impact of time-step size on the numerical dispersion, and as a consequence, significantly reduces numerical dispersion error for any time-step size.

Numerical Stability and Dispersion Analysis of the Precise-Integration Time-Domain Method in Lossy Media

In this paper, both the numerical stability condition and dispersion relation of the precise-integration time-domain (PITD) method in lossy media are presented. It is found that the time step size of the PITD method is limited by both the spatial step size of the PITD method and the ratio of permittivity to conductivity. In numerical dispersion investigations, it is shown that: the numerical loss error of the PITD method is always positive; the numerical phase error of the PITD method can be positive or negative; the numerical loss and phase errors can be made nearly independent of the time step size; and as the spatial step size decreases, the amplitudes of the numerical loss and phase errors decrease. In good conductors, the numerical phase velocity of the PITD method is closer to the physical value as compared with the finite-difference time-domain method. The numerical phase anisotropy of the PITD method can be positive or negative. The numerical anisotropies of the PITD method in the 3-D case are usually larger than those in the 2-D case. There is a conductivity giving zero numerical phase anisotropy. These theoretical observations are confirmed by numerical experiments.

A Low-Memory-Requirement Realization of Precise Integration Time Domain Method Using a Leapfrog Scheme

In this letter, the leapfrog scheme-based precise integration time domain (L-PITD) method is proposed to decrease the memory requirements of the precise integration time domain (PITD) method. In the L-PITD method, the electromagnetic field components are divided into two groups so as to split up a 3-D problem to a set of 2-D problems. And then a leapfrog scheme is proposed to solve these 2-D problems by using the precise integration technique. The numerical results confirm that the L-PITD method not only is stable with the time step size much larger than the Courant-Friedrich-Levy limit of the finite difference time domain method, but also has significant advantages over the PITD method with respect to the execution time and the memory requirements.

A Low-Dispersion Realization of Precise Integration Time-Domain Method Using a Fourth-Order Accurate Finite Difference Scheme

A modified precise integration time-domain (PITD) method, called PITD(4) algorithm, is presented in order to mitigate the numerical dispersion errors of a recently proposed PITD method. The PITD(4) method is based on both the fourth-order accurate finite-difference scheme and the precise integration technique. Both the stability condition and the numerical dispersion relations of the PITD(4) method are derived analytically and the effects of spatial and time steps on the numerical dispersion are investigated in detail. It is found that with the precise integration technique, the stability condition of the PITD(4) method is much larger than the Courant-Friedrich-levy (CFL) stability condition of the conventional finite difference time domain; with the fourth-order accurate finite-difference scheme, the numerical dispersion errors of the PITD(4) method are much less than that of the PITD methods. Numerical examples are presented to validate the accuracy and the effectiveness of the PITD(4) method, and to verify our analysis of the numerical dispersion characteristics of the PITD(4) method.

A Low Dispersion Precise Integration Time Domain Method Based on Wavelet Galerkin Scheme

To decrease the dispersion error of the precise integration time domain (PITD) method, a new algorithm named the wavelet Galerkin scheme-based PITD (WG-PITD) method is proposed in this letter. The novel method is based on both the wavelet Galerkin scheme and the precise integration technique. The dispersion relation of the WG-PITD method is derived analytically. It is found that the dispersion error of the WG-PITD method is smaller than that of the PITD method, and can be made nearly independent of the time step size. The numerical results confirm the advantages of the WG-PITD method over the PITD method with respect to the memory requirements and the execution time.

A 3-D unconditionally stable precise integration time domain method for the numerical solutions of Maxwell's equations in circular cylindrical coordinates

A 3-D unconditionally stable precise integration time domain method for the numerical solutions of Maxwell's equations in circular cylindrical coordinates

Electromagnetic Closed-Surface Criterion for the 3-D Precise Integration Time-Domain Method for Solving Maxwell's Equations

In order to derive the linear relations for solving the singular matrix problem during the recurrence process of the precise integration time-domain method, an electromagnetic closed-surface criterion is presented in orthogonal curvilinear coordinates. Such a criterion leads to the versatile method by which the spatial distributions of the linearly dependent field components are determined. It makes the complicated derivation for linear relation, usually depending on experience, regular, and therefore improves the general-purpose property of the method. With such a criterion, linear relations have been obtained in the computational domain, on the absorbing boundaries and on the edges where boundaries cross in orthogonal curvilinear coordinates. Numerical examples indicate the validity of the criterion.

A 3-D precise integration time-domain method without the restraints of the courant-friedrich-levy stability condition for the numerical solution of Maxwell's equations

In this paper, a new three-dimensional time-domain method for solving vector Maxwell's equations, called the precise-integration time-domain (PITD) algorithm, is proposed in order to eliminate the Courant-Friedrich-Levy (CFL) condition restraint. The new algorithm is based on the precise-integration technique. It is shown that this method is quite stable even when the CFL condition is not satisfied. Although the memory requirement of the PITD method is much larger than that of the finite-difference time-domain (FDTD) method, this new algorithm is very appealing since the time step used in the simulation is no longer restricted by stability. As a result, computation speed can be improved. Therefore, if the minimum cell size in the computational domain is required to be much smaller than the wavelength, this new algorithm is more efficient than the FDTD scheme. Theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness and efficiency of the method. It is found that the accuracy of the PITD is independent of the time-step size.