Analysis of spatial high-order finite difference methods for Maxwell's equations in dispersive media

Bokil, Vrushali A.; Gibson, Nathan L.

doi:10.1093/imanum/drr001

Cited by 39 publications

(17 citation statements)

References 24 publications

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“…This and similar types of analysis can be found in [1,3,5,10] and in references therein. First, substitute the plane wave ansatz (4.2) into the discrete scheme (3.13) and cancel the time-dependent exponential term present on both sides: (4.13) We then pose the discrete dispersion relationship as a global eigenvalue problem…”

Section: The Discrete Dispersion Relationsupporting

confidence: 63%

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Dispersion reducing methods for edge discretizations of the electric vector wave equation

Bokil

Gibson

Gyrya

et al. 2015

Journal of Computational Physics

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We present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time-domain electric vector wave equation on square meshes based on the mimetic finite difference (MFD) method. We compare this strategy, called M-adaptation, to two other discretizations, also based on square meshes. One is the lowest order Nédélec edge element discretization. The other is a modified quadrature approach (GY-adaptation) proposed by Guddati and Yue for the acoustic wave equation in two dimensions. All three discrete methods use the same edge-based degrees of freedom, while the temporal discretization is performed using the standard explicit Leapfrog scheme. To obtain efficient and explicit time stepping methods, the three schemes are further mass lumped. We perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error. Our results indicate that the method produced by GY-adaptation and the Nédélec method are both second order accurate for numerical dispersion, but differ in the order of their numerical anisotropy (fourth order, versus second order, respectively). The result of M-adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy. Numerical simulations are provided that illustrate the theoretical results.

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Section: The Discrete Dispersion Relationsupporting

confidence: 63%

“…Here the scaling for the functions v 3 and v 4 is selected for convenience, to make the columns T j of the transformation matrix T , defined by the d.o.f. of v j , independent of the dimensions of the element:…”

Section: Nédélec's Edge Elementsmentioning

confidence: 99%

Dispersion reducing methods for edge discretizations of the electric vector wave equation

Bokil

Gibson

Gyrya

et al. 2015

Journal of Computational Physics

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“…However, when modeling high loss and/or large dielectric constants these methods become unstable. To improve the stability and calculation accuracy for high dielectric dispersive media, the high-order finite difference scheme in space domain and exponential time differencing algorithm in time domain have been used to model electromagnetic wave propagation [4][5][6][7][8][9][10][11]. In [4], a fourth-order accurate in space and second-order accurate in time FDTD scheme are presented to modeling wave propagation in lossy dispersive media.…”

Section: Introductionmentioning

confidence: 99%

“…In [4], a fourth-order accurate in space and second-order accurate in time FDTD scheme are presented to modeling wave propagation in lossy dispersive media. In [5], the stability property and numerical dispersion relation for high-order FDTD scheme with a Debye or Lorentz model are analyzed. In [4] and [5], the high-order scheme is used in space, and numerical dissipation is strongly dependent on the temporal resolution.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], the stability property and numerical dispersion relation for high-order FDTD scheme with a Debye or Lorentz model are analyzed. In [4] and [5], the high-order scheme is used in space, and numerical dissipation is strongly dependent on the temporal resolution. In [6], the ETD scheme for FDTD is proposed to model electromagnetic wave propagation in an isotropic homogeneous lossy dielectric with electric and magnetic conductivities σ and σ * , respectively.…”

Section: Introductionmentioning

confidence: 99%

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High-Order FDTD With Exponential Time Differencing Algorithm for Modeling Wave Propagation in Debye Dispersive Materials

Chen¹,

Tang²

2018

PIER Letters

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Abstract-A high-order (HO) finite-difference time-domain (FDTD) method with exponential time differencing (ETD) algorithm is proposed to model electromagnetic wave propagation in Debye dispersive material in this paper. The proposed method introduces an auxiliary difference equation (ADE) technique which establishes the relationship between the electric displacement vector and electric field intensity with a differential equation in Debye dispersive media. The ETD algorithm is applied to the displacement vector and auxiliary difference variable in time domain, and the fourth-order centraldifference discretization is used in space domain. One example with plane wave propagation in a Debye dispersive media is calculated. Compared with the conventional ETD-FDTD method, the results from our proposed method show its accuracy and efficiency for Debye dispersive media simulation.

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An improved two‐dimensional (2,4) finite‐difference time‐domain method for Lorentz dispersive media

Zygiridis,

Amanatiadis,

Kantartzis

2024

Int J Numerical Modelling

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The credible solution of discretized Maxwell's equations in spaces occupied by Lorentz dispersive media is the main subject of this work. Specifically, we introduce a finite‐difference time‐domain (FDTD) algorithm with a typical (2,4) structure that features dispersion‐relation‐preserving characteristics and produces reduced numerical errors in two‐dimensional electromagnetic simulations, compared to the standard approach with similar computational requirements. We consider the case of dispersive media with non‐vanishing absorption coefficients and investigate different options for the suitable modification of the spatial approximations, so that the accomplished accuracy is optimized for a given computational overhead. The properties of the proposed FDTD technique are thoroughly examined, both theoretically and in numerical tests, and the performance upgrade compared with the conventional solution is assessed.

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Analysis of spatial high-order finite difference methods for Maxwell's equations in dispersive media

Cited by 39 publications

References 24 publications

Dispersion reducing methods for edge discretizations of the electric vector wave equation

Dispersion reducing methods for edge discretizations of the electric vector wave equation

High-Order FDTD With Exponential Time Differencing Algorithm for Modeling Wave Propagation in Debye Dispersive Materials

An improved two‐dimensional (2,4) finite‐difference time‐domain method for Lorentz dispersive media

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