Equations In Dispersive Media Research Articles

Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Convergence analysis of operator splitting methods for Maxwell’s equations in dispersive media of Debye type

Soliton structures and dynamical behaviors for the integrable system of Drinfel’d–Sokolov–Wilson equations in dispersive media

This study is used to investigate the exact explicit solutions and dynamical behaviors of the (1+1)-dimensional integrable system of Drinfel’d–Sokolov–Wilson equations in dispersive media. Firstly, the unified method is implemented to find explicit solutions in polynomial and rational forms. These solutions include rational, dark and bright soliton structures. After that, the planar dynamical system of the considered model is obtained using the Galilean transformation. The phase portraits of the bifurcations are drawn from the planar dynamical system for different physical parametric values using the fourth-order Runge–Kutta method. Further, an external force is imposed on the dynamical system to observe the phase portraits of chaotic trajectories. These tracks are outlined for different values of the strength and frequency of the external force acting on the dynamical system. The outcomes show that a given model has chaotic behvior as its solution becomes disordered by taking small variation in the strength and frequency of the external force. The results are new and can be helpful in investigating the dynamical behaviors of the other nonlinear physical models arising in optics, bio-mathematics and so many other fields of science.

Steeping and dispersive effects analysis of a couple of long-wave equations in dispersive media

Dispersive wave propagations are described by nonlinear models, such as regularized long waves and modified regularized long waves, where nonlinearity and dispersion are important aspects of wave evolution to model long-wave propagation in dispersive media with small amplitudes. In this article, solitary wave solutions to the formerly indicated nonlinear equations are computed as exponential, hyperbolic, and trigonometric structures and their integration by balancing the steeping and dispersion terms of the highest order, and different classes of definitive localized coherent structures with their interaction properties, such as soliton solutions, are extracted by assigning appropriate functions. We have also investigated the impact of steeping and wave spread on waveforms. The wave solutions are assessed using the enhanced modified simple equation approach, which organizes the soliton solutions based on the features of the dispersive interconnections with the nonlinear cubic and quadratic characters that exist in the equations. The consistent 3D and 2D soliton configurations show that the wave profile is modulated by the wave velocity function and the parameters associated with it, and most of the modulation is due to linear effects.

A reduced-order DG formulation based on POD method for the time-domain Maxwell’s equations in dispersive media

In this work, a proper orthogonal decomposition (POD) method is applied to time-domain Maxwell’s equations coupled to a Drude dispersion model, which are discretized in space by a discontinuous Galerkin (DG) method. An auxiliary differential equation (ADE) method is used to represent the constitutive relation for the dispersive medium. A POD–DGTD formulation with lower dimension and sufficiently high accuracy is established, together with the description of the POD reduced-order basis, its construction from a snapshot set, and its application to the solution of the time-domain Maxwell’s equations. The overall goal is to reduce the computational time while maintaining an acceptable level of accuracy, in order to obtain an efficient time-domain solver to be used as a starting-point for an optimization strategy. We provide results from numerical experiments for two-dimensional problems that illustrate the capabilities of the proposed POD–DGTD formulation and assess its efficiency.

Space-time discontinuous galerkin method for maxwell equations in dispersive media

In this paper, a unified model for time-dependent Maxwell equations in dispersive media is considered. The space-time DG method developed in [29] is applied to solve the underlying problem. Unconditional L2-stability and error estimate of order O(τr+1+hk+1/2) are obtained when polynomials of degree at most r and k are used for the temporal discretization and spatial discretization respectively. 2-D and 3-D numerical examples are given to validate the theoretical results. Moreover, numerical results show an ultra-convergence of order 2r+1 in temporal variable t.

Operator splitting methods for Maxwell’s equations in dispersive media with orientational polarization

We present two operator splitting schemes for the numerical simulation of Maxwell’s equations in dispersive media of Debye type that exhibit orientational polarization (the Maxwell–Debye model). The splitting schemes separate the mechanisms of wave propagation and polarization to create simpler sub-steps that are easier to implement. In addition, dimensional splitting is used to propagate waves in different axial directions. We present a sequential operator splitting scheme and its symmetrized version for the Maxwell–Debye system in two dimensions. The splitting schemes are discretized using implicit finite difference methods that lead to unconditionally stable schemes. We prove that the fully discretized sequential scheme is a first order time perturbation, and the symmetrized scheme is a second order time perturbation of the Crank–Nicolson scheme for discretizing the Maxwell–Debye model. Numerical examples are presented that illustrate our theoretical results.

Convergence of a discontinuous Galerkin scheme for the mixed time-domain Maxwell's equations in dispersive media

This work is about the numerical solution of the time-domain Maxwell's equations in dispersive propagation media by a discontinuous Galerkin time-domain method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of differential equations is solved using a centred-flux discontinuous Galerkin formulation for the discretization in space and a second-order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an auxiliary differential equation approach similar to that which is adopted in the finite difference time-domain method. Stability estimates are derived through energy considerations and convergence is proved for both the semidiscrete and the fully discrete schemes.

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

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

Theory of coupling in dispersive photonic systems

Based on a recently developed Hamiltonian form of Maxwell's equations for dispersive media [A. Raman and S. Fan, Phys. Rev. Lett. 104, 087401 (2010)], we established a general theoretical framework to study the coupling problems in dispersive photonic systems such as plasmonic materials. Our theory can quantitatively determine the coupling strengths between two dispersive photonic resonators based on ab initio full-wave calculations on realistic structures. The theory is validated by comparing with known rigorous results, where the crucial role of dispersion correction is identified. As an application, we studied the coupled localized surface plasmons in a double-nanorod system. The obtained results agree well with full-wave simulations in most cases, and are helpful to study the ``dark modes'' which are difficult to detect from conventional simulations.

Interior penalty DG methods for Maxwell’s equations in dispersive media

In this paper, we develop a fully-discrete interior penalty discontinuous Galerkin method for solving the time-dependent Maxwell’s equations in dispersive media. The model is described by a vector integral–differential equation. Our scheme is proved to be unconditionally stable and achieve optimal error estimates in both L 2 norm and energy norm. The scheme is implemented and numerical results supporting our analysis are presented.

Unified Analysis of Leap-Frog Methods for Solving Time-Domain Maxwell’s Equations in Dispersive Media

In this paper, we consider the time dependent Maxwell's equations resulting from dispersive medium models. First, the stability and Gauss's law are proved for all three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Then leap-frog mixed finite element methods are developed for these three models. Optimal error estimates are proved for all three models solved by the lowest-order Raviart-Thomas-Nedelec spaces. Extensions to multiple pole dispersive media are presented also. Numerical results confirming the analysis are presented.

Error analysis of a discontinuous Galerkin method for Maxwell equations in dispersive media

A discontinuous Galerkin method for the numerical approximation of time-dependent Maxwell equations in three different dispersive media is introduced. Both the L 2-stability and error estimate of the DG method are discussed in detail. We show that the proposed method has an accuracy of O ( h k + 1 2 ) under the L 2-norm when polynomials of degree k in space are used. Furthermore, numerical experiments are provided to justify our theoretical analysis.

Unified Analysis of Time Domain Mixed Finite Element Methods for Maxwell's Equations in Dispersive Media

In this paper, we consider the time dependent Maxwell’s equations when dispersive media are involved. The Crank-Nicolson mixed finite element methods are developed for three most popular dispersive medium models: the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Optimal error estimates are proved for all three models solved by the Raviart-Thomas-Nedelec spaces. Extensions to multiple pole dispersive media are presented also. Mathematics subject classification: 65N30, 35L15, 78-08.

Solution of time-convolutionary Maxwell’s equations using parameter-dependent Krylov subspace reduction

We suggest a new algorithm for the solution of the time domain Maxwell equations in dispersive media. After spacial discretization we obtain a large system of time-convolution equations. Then this system is projected onto a small subspace consisting of the Laplace domain solutions for a preselected set of Laplace parameters. This approach is a generalization of the rational Krylov subspace approach for the solution of non-dispersive Maxwell’s systems. We show that the projected system preserves such properties of the initial system as stability and passivity. As an example we consider the 3D quasistationary induced polarization problem with the Cole–Cole conductivity model important for geophysical oil exploration. Our numerical experiments show that the introduction of the induced polarization does not have significant effect on convergence.

Interior Penalty Discontinuous Galerkin Method for Maxwell’s Equations in Cold Plasma

In this paper, we develop an interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in cold plasma. Both semi and fully discrete DG schemes are constructed, and optimal error estimates in the energy norm are proved. To our best knowledge, this is the first error analysis carried out for the DG method for Maxwell's equations in dispersive media.

Finite element study of time‐dependent Maxwell's equations in dispersive media

AbstractIn this article, we consider the time‐dependent Maxwell's equations in a bounded domain when dispersive media are involved. The Crank‐Nicolson scheme is developed to approximate the electric field equation by Nedelec edge elements and is proved to be optimal convergent in energy norm. The analysis is carried out for Debye medium, but the same results hold true for other dispersive media such as plasma and Lorentz medium. Furthermore, our analysis extends straightforward to cases when a dispersive medium and a simple medium (such as air) are coupled. Mathematics Subject Classification (2000): 65N30, 35L15, 78‐08. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Superconvergence analysis for Maxwell's equations in dispersive media

In this paper, we consider the time dependent Maxwell’s equations in dispersive media on a bounded three-dimensional domain. Global superconvergence is obtained for semi-discrete mixed finite element methods for three most popular dispersive media models: the isotropic cold plasma, the one-pole Debye medium, and the two-pole Lorentz medium. Global superconvergence for a standard finite element method is also presented. To our best knowledge, this is the first superconvergence analysis obtained for Maxwell’s equations when dispersive media are involved.

Error analysis of fully discrete mixed finite element schemes for 3-D Maxwell’s equations in dispersive media

We consider the time dependent Maxwell’s equations in dispersive media in a bounded three-dimensional (3-D) domain. Fully discrete mixed finite element methods are developed for three most popular dispersive media models: i.e., the isotropic cold plasma, the one-pole Debye medium and the two-pole Lorentz medium. Optimal error estimates are proved for those three dispersive media under the assumption that the solutions are smooth enough. Numerical results justifying our analysis are provided. We believe that this is the first error analysis carried out for fully discrete mixed finite element methods for solving 3-D Maxwell’s equations in dispersive media.

Analysis of a time-domain finite element method for 3-D Maxwell’s equations in dispersive media

We consider the time dependent Maxwell’s equations in dispersive media on a bounded domain in three-dimensional space. A fully discrete finite element scheme is developed to approximate the electric field equation derived from the Maxwell’s equations. Optimal energy-norm error estimates are proved for Nédélec curl-conforming edge elements. This is the first finite element error analysis for Maxwell’s equations in dispersive media.

Error analysis of finite element methods for 3-D Maxwell's equations in dispersive media

We consider the time dependent Maxwell's equations in dispersive media on a bounded three-dimensional domain. A semi-discrete standard finite element method and a semi-discrete mixed finite element method are developed and error estimates are provided. We believe this is the first finite element error analysis carried out for Maxwell's equations in dispersive media.