scholarly journals DEVELOPMENT OF A SYMPLECTIC SCHEME WITH OPTIMIZED NUMERICAL DISPERSION-RELATION EQUATION TO SOLVE MAXWELL'S EQUATIONS IN DISPERSIVE MEDIA

2012 ◽  
Vol 132 ◽  
pp. 517-549 ◽  
Author(s):  
Tony W. H. Sheu ◽  
Rih Yang Chung ◽  
Jia-Han Li
Keyword(s):  
Dispersion Relation ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Numerical Dispersion ◽  
Dispersive Media ◽  
Symplectic Scheme

Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell’s Equations

2013 ◽  
Vol 13 (4) ◽  
pp. 1107-1133 ◽  
Author(s):  
Tony W. H. Sheu ◽  
L. Y. Liang ◽  
J. H. Li
Keyword(s):  
Dispersion Relation ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Angular Frequency ◽  
Numerical Dispersion ◽  
Local Conservation ◽  
Symplectic Scheme ◽  
Electric And Magnetic Fields ◽  
The Difference ◽  
Explicit Finite Difference

AbstractIn this paper an explicit finite-difference time-domain scheme for solving the Maxwell’s equations in non-staggered grids is presented. The proposed scheme for solving the Faraday’s and Ampere’s equations in a theoretical manner is aimed to preserve discrete zero-divergence for the electric and magnetic fields. The inherent local conservation laws in Maxwell’s equations are also preserved discretely all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme. The remaining spatial derivative terms in the semi-discretized Faraday’s and Ampere’s equations are then discretized to provide an accurate mathematical dispersion relation equation that governs the numerical angular frequency and the wavenumbers in two space dimensions. To achieve the goal of getting the best dispersive characteristics, we propose a fourth-order accurate space centered scheme which minimizes the difference between the exact and numerical dispersion relation equations. Through the computational exercises, the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions.

A simplified Lax‐Wendroff correction for staggered‐grid FDTD modeling of electromagnetic wave propagation in frequency‐dependent media

Geophysics ◽  
1999 ◽  
Vol 64 (5) ◽  
pp. 1369-1377 ◽  
Author(s):  
Tim Bergmann ◽  
Joakim O. Blanch ◽  
Johan O. A. Robertsson ◽  
Klaus Holliger
Keyword(s):  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Numerical Dispersion ◽  
Staggered Grid ◽  
Dispersive Media ◽  
Frequency Dependent ◽  
Elegant Method ◽  
Constitutive Parameters

The Lax‐Wendroff correction is an elegant method for increasing the accuracy and computational efficiency of finite‐difference time‐domain (FDTD) solutions of hyperbolic problems. However, the conventional approach leads to implicit solutions for staggered‐grid FDTD approximations of Maxwell’s equations with frequency‐dependent constitutive parameters. To overcome this problem, we propose an approximation that only retains the purely acoustic, i.e., lossless, terms of the Lax‐Wendroff correction. This modified Lax‐Wendroff correction is applied to an O(2, 4) accurate staggered‐grid FDTD approximation of Maxwell’s equations in the radar frequency range (≈10 MHz–10 GHz). The resulting pseudo-O(4, 4) scheme is explicit and computationally efficient and exhibits all the major numerical characteristics of an O(4, 4) accurate FDTD scheme, even for strongly attenuating and dispersive media. The numerical properties of our approach are constrained by classical numerical dispersion and von Neumann‐Routh stability analyses, verified by comparisons with pertinent 1-D analytical solutions and illustrated through 2-D simulations in a variety of surficial materials. Compared to the O(2, 4) scheme, the pseudo-O(4, 4) scheme requires 64% fewer grid points in two dimensions and 78% in three dimensions to achieve the same level of numerical accuracy, which results in large savings in core memory.

Multi-Symplectic Wavelet Collocation Method for Maxwell’s Equations

2011 ◽  
Vol 3 (6) ◽  
pp. 663-688 ◽  
Author(s):  
Huajun Zhu ◽  
Songhe Song ◽  
Yaming Chen
Keyword(s):  
Collocation Method ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Time Integration ◽  
Three Dimensional ◽  
Numerical Dispersion ◽  
Splitting Scheme ◽  
Spatial Accuracy ◽  
Symplectic Scheme ◽  
Wavelet Collocation Method

AbstractIn this paper, we develop a multi-symplectic wavelet collocation method for three-dimensional (3-D) Maxwell’s equations. For the multi-symplectic formulation of the equations, wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration. Theoretical analysis shows that the proposed method is multi-symplectic, unconditionally stable and energy-preserving under periodic boundary conditions. The numerical dispersion relation is investigated. Combined with splitting scheme, an explicit splitting symplectic wavelet collocation method is also constructed. Numerical experiments illustrate that the proposed methods are efficient, have high spatial accuracy and can preserve energy conservation laws exactly.

Kinetic theory of surface wave propagation along a hot plasma column

1976 ◽  
Vol 16 (1) ◽  
pp. 47-55 ◽  
Author(s):  
V. Atanassov ◽  
I. Zhelyazkov ◽  
A. Shivarova ◽  
Zh. Genchev
Keyword(s):  
Dispersion Relation ◽  
Plasma Column ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Displacement Vector ◽  
Electric Displacement ◽  
Wave Dispersion ◽  
Axially Symmetric ◽  
Surface Wave Propagation ◽  
Hot Plasma

In this paper we propose an exact solution of Vlasov and Maxwell's equations for a bounded hot plasma in order to derive the dispersion relation of the axially-symmetric surface waves propagating along a plasma column. Assuming specular reflexion of plasma particles from the boundary, expressions for the components of the electric displacement vector are obtained on the basis of the Vlasov equation. Their substitution in Maxwell's equations, neglecting the spatial dispersion in the transverse plasma dielectric function, allows us to determine the plasma impedance. The equating of plasma and dielectric impedances gives the wave dispersion relation which, in different limiting cases, coincides with the well-known results.

3-D simulation of GPR surveys over pipes in dispersive soils

Geophysics ◽  
2000 ◽  
Vol 65 (5) ◽  
pp. 1560-1568 ◽  
Author(s):  
Tsili Wang ◽  
Michael L. Oristaglio
Keyword(s):  
Electrical Properties ◽  
Time Domain ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Displacement Vector ◽  
Electric Displacement ◽  
Field Vector ◽  
Dispersive Media ◽  
Dispersive Soils ◽  
Dispersive Soil

The finite‐difference time‐domain method is adapted to simulate radar surveys of objects buried in dispersive soils whose complex permittivity depends on frequency. The method treats dispersion through the constitutive relation between the electric field vector and the electric displacement vector, which is a convolution in the time domain. This convolution is updated recursively, along with Maxwell’s equations, after approximating the dispersion with a Debye (exponential) relaxation model. A novel feature of our work is the inclusion of dispersion in the perfectly‐matched layer formulation of Maxwell’s equations, which gives an absorbing boundary condition for dispersive media. We simulate 200-MHz ground‐penetrating radar surveys over metallic and plastic pipes buried at a depth of 2 m in soils whose electrical properties model are those of clay loams of different moisture contents. Radar reflections modeled for pipes in dispersive soil differ from those for pipes in soils whose electrical properties are constant (at the values of dispersive soil at the central frequency of the radar pulse). Because the permittivity decreases at higher frequencies in the soils modeled, energy in the reflections shifts toward the front of the waveform, and the amplitudes of trailing lobes in the waveform are suppressed. The effects are subtle, but become more pronounced in models of soils with 10% moisture content by weight.

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