Alternating-direction Implicit Finite-difference Time-domain Scheme Research Articles

An ADI-FDTD Formulation With Modified Lorentz Dispersion for the Study of Plasmonic Structures

This letter presents an alternating-direction implicit finite-difference time-domain scheme for the efficient study of plasmonic systems. The material dispersion is described by generalized modified Lorentzian terms and it is implemented via the auxiliary differential equations technique employing an order reduction. The computational domain is backed by a properly designed convolution perfectly matched layer. The efficiency of the proposed method is validated in benchmark examples and its unconditional stability is evidenced by the Fourier method.

Optimal error estimates and modified energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell’s equations

This paper is concerned with the optimal error estimates and energy conservation properties of the alternating direction implicit finite-difference time-domain (ADI-FDTD) method which is a popular scheme for solving the 3D Maxwell’s equations. Precisely, for the case with a perfectly electric conducting (PEC) boundary condition we establish the optimal second-order error estimates in both space and time in the discrete H1-norm for the ADI-FDTD scheme, and prove the approximate divergence preserving property that if the divergence of the initial electric and magnetic fields are zero, then the discrete L2-norm of the discrete divergence of the ADI-FDTD solution is approximately zero with the second-order accuracy in both space and time. The key ingredient is two new discrete modified energy norms which are second-order in time perturbations of two new energy conservation laws for the Maxwell’s equations introduced in this paper. Furthermore, we prove that, in addition to two known discrete modified energy identities which are second-order in time perturbations of two known energy conservation laws, the ADI-FDTD scheme also satisfies two new discrete modified energy identities which are second-order in time perturbations of the two new energy conservation laws. This means that the ADI-FDTD scheme is unconditionally stable under the four discrete modified energy norms. Experimental results which confirm the theoretical results are presented.

Split-step finite difference time domain algorithm for double negative meta-materials

Unconditionally stable finite difference time domain (FDTD) formulation is presented for modelling electromagnetic wave propagation in open region double-negative meta-material domains. The formulation is based on incorporating the bilinear transformation technique and the stretched coordinate perfectly matched layer into the split-step FDTD algorithm. Numerical examples carried out in one- and two-dimensional domains show that the proposed formulation not only maintains the accuracy level of the conventional alternating direction implicit FDTD scheme but also reduces the required computational time by 21%.

Three-dimensional hybrid implicit–explicit finite-difference time-domain method in the cylindrical coordinate system

A novel hybrid implicit–explicit (HIE) finite-difference time-domain (FDTD) method, which is extremely useful for problems with very fine structures along the ϕ-direction in cylindrical coordinate system, is presented. This method has higher computation efficiency than conventional cylindrical FDTD methods, because the time step in this method is only determined by the space discretisations in the radial and vertical directions. The numerical stability of the proposed HIE–FDTD method is presented analytically. Compared with the cylindrical alternating-direction implicit (ADI)–FDTD method, this HIE–FDTD method has higher accuracy, especially for larger time step size. At each time step, the HIE–FDTD method requires the solution of two tridiagonal matrices and four explicit updates. While maintaining the same size of time step, the central processing unit (CPU) time for this weakly conditionally stable FDTD method can be reduced to about 3/5 of that for the ADI–FDTD scheme. The numerical performance of the proposed HIE–FDTD over the conventional cylindrical FDTD method and cylindrical ADI–FDTD method is demonstrated through numerical examples.

Symmetric source implementation for the ADI-FDTD method

The alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method has been introduced to overcome the Courant limit present in the standard FDTD scheme. However, the ADI-FDTD scheme may produce small asymmetries in the field distributions according to the source implementation. In this communication, we compare different source implementation strategies and show that ADI-FDTD scheme has no asymmetry errors up to the numerical noise level, and can be implemented most accurately if the excitation is directly incorporated within the tridiagonal matrix and if the time discretization of the source is done appropriately within each full time step.

Accurate implementation of current sources in the ADI-FDTD scheme

This letter presents a new method of implementing current sources in the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method. The current source condition is derived systematically, starting with the fully implicit Crank-Nicolson (CN) FDTD scheme for Maxwell's curl equations and building the ADI-FDTD scheme as a second-order-in-time perturbation of the CN-FDTD scheme. We demonstrate that the accuracy of this new source condition is superior to previously proposed techniques.

General Formulation of Unconditionally Stable ADI–FDTD Method in Linear Dispersive Media

The unconditionally stable alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method is used to model wave propagation in dispersive media. A formulation is presented by introducing the Z-transform method into the ADI-FDTD scheme to handle the frequency-dependent features of the media. This formulation is applicable to arbitrary dispersive media, and can be easily coded. Numerical results are compared to those based on the conventional FDTD method to show the efficiency of the proposed method.

On the accuracy of the ADI-FDTD method

We present an analytical study of the alternating-direction implicit finite-difference time-domain (ADI-FDTD) method for solving time-varying Maxwell's equations and compare its accuracy with that of the Crank-Nicolson (CN) and Yee FDTD schemes. The closed form of the truncation error is obtained for two and three dimensions. The dependence of the truncation error on the square of the time step multiplied by the spatial derivatives of the fields is found to be a unique feature of the ADI-FDTD scheme. We illustrate the limitation on accuracy imposed by these truncation error terms by simulating a simple parallel-plate structure excited by a low-frequency voltage source. Excellent agreement is obtained between field data computed with the implicit CN scheme using time steps greatly exceeding the Courant limit and field data computed with the explicit Yee scheme. Such results are not obtained with ADI-FDTD.