High-order FDTD Scheme Research Articles

High-Order FDTD Schemes for Maxwell’s Interface Problems with Discontinuous Coefficients and Complex Interfaces Based on the Correction Function Method

We propose high-order FDTD schemes based on the Correction Function Method (CFM) (Marques et al. in J Comput Phys 230:7567–7597, 2011) for Maxwell’s interface problems with discontinuous coefficients and complex interfaces. The key idea of the CFM is to model the correction function near an interface to retain the order of a finite difference approximation. To do so, we solve a system of PDEs based on the original problem by minimizing an energy functional. The CFM is applied to the standard Yee scheme and a fourth-order FDTD scheme. The proposed CFM-FDTD schemes are verified in 2-D using the transverse magnetic (\(\hbox {TM}_z\)) mode. Numerical examples include scattering of magnetic and non-magnetic dielectrics, and problems with manufactured solutions using various complex interfaces and discontinuous piecewise varying coefficients. Long-time simulations are also performed to investigate the stability of CFM-FDTD schemes. The proposed CFM-FDTD schemes achieve up to fourth-order convergence in \(L^2\)-norm and provide approximations devoid of spurious oscillations.

Treatment of Complex Interfaces for Maxwell’s Equations with Continuous Coefficients Using the Correction Function Method

We propose a high-order FDTD scheme based on the correction function method (CFM) to treat interfaces with complex geometry without significantly increasing the complexity of the numerical approach for constant coefficients. Correction functions are modeled by a system of PDEs based on Maxwell’s equations with interface conditions. To be able to compute approximations of correction functions, a functional that is a square measure of the error associated with the correction functions’ system of PDEs is minimized in a divergence-free discrete functional space. Afterward, approximations of correction functions are used to correct a FDTD scheme in the vicinity of an interface where it is needed. We perform a perturbation analysis on the correction functions’ system of PDEs. The discrete divergence constraint and the consistency of resulting schemes are studied. Numerical experiments are performed for problems with different geometries of the interface. A second-order convergence is obtained for a second-order FDTD scheme corrected using the CFM. High-order convergence is obtained with a corrected fourth-order FDTD scheme. The discontinuities within solutions are accurately captured without spurious oscillations.

Constructing nonstandard and high order FDTD schemes in cylindrical coordinates using spectral domain and modified equation methodologies – 1-D case

One of the major drawbacks of the original Yee’s algorithm is “staircasing.” In an attempt to overcome this flaw, especially when trying to solve electromagnetic problems involving cylindrical geometries, an FDTD scheme was formulated in cylindrical coordinates. The vast majority of works published describe an algorithm with a (2, 2) order of accuracy. In this work, we use a recently introduced methodology, that of constructing finite difference schemes in the spectral domain, combined with the well-established one known as the “modified equation” to analyze existing cylindrical FDTD schemes and synthesize new nonstandard and high-order ones.

Nondiagonally anisotropic PML: a generalized unsplit wide-angle absorber for the treatment of the near-grazing effect in FDTD meshes

A new generalization of the PML with wide-angle absorption is presented, for the efficient truncation of FDTD lattices. The proposed unsplit-field layer uses a nondiagonal symmetric tensor anisotropy and via an appropriate parameter selection achieves notable attenuation rates in the case of near-grazing angles. Hence, it can be placed much closer to large structures. Improved accuracy and lower dispersion errors are attained via higher-order FDTD schemes with regular and non-Cartesian lattices constructed by hexagonal or tetradecahedral cells. Finally, Ramahi ABC's are invoked for the absorber's termination. Numerical results, addressing wave attenuation at grazing incidence, prove the efficiency of the proposed PML.

Fully nonorthogonal higher-order FDTD schemes for the systematic development of 3-D PML's in general curvilinear coordinates

The efficient construction of reflectionless PML's in 3-D curvilinear coordinates via a new higher-order FDTD methodology, is presented in this paper. By accurately treating the div-curl problem, the technique introduces a higher-order rendition of the covariant/contravariant vector theory with generalized conventional and nonstandard schemes. Moreover, a mesh expansion algorithm decreases the absorbers' thickness. In the time domain, the four-stage Runge-Kutta integrator is also invoked, while the wider spatial stencils are effectively limited by self-adaptive compact operators. Numerical verification indicates that the novel PML's offer a serious suppression of dispersion errors and significant savings in the computational burden.