Time-dependent Maxwell Equations Research Articles

Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Dependent First Order Maxwell's Equations

An interior penalty discontinuous Galerkin method is described with a conformal perfectly matched layer (PML) for solving the two first-order Maxwell's equations in the time domain. Both central and upwind fluxes are studied in this work. In both cases, the proposed method is explicit and conditionally stable. Additionally, a local time-stepping strategy is applied to increase efficiency and reduce the computational time. Finally, numerical examples are presented to validate the method.

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.

Explicit local time-stepping methods for Maxwell’s equations

Explicit local time-stepping methods are derived for time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretisation in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fully explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leap-frog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes.

The Discontinuous Galerkin Time-Domain method for Maxwell’s equations with anisotropic materials

The Discontinuous Galerkin method is an accurate and efficient way to numerically solve the time-dependent Maxwell equations. In this paper, we extend the basic, two-dimensional formulation for isotropic materials to allow anisotropic permittivity tensors. Using a reference system with an analytical solution, we demonstrate that our extensions do not alter the superior convergence characteristics of the fundamental algorithm. We further apply our method to cylindrical invisibility cloaks to investigate the performance which can be achieved in experiments.

Three-Dimensional Electromagnetic Scattering Analysis Using Constrained Interpolation Profile Method

A characteristic-based constrained interpolation profile (CIP) method for solving three-dimensional, time-dependent Maxwell's equations is successfully developed. It is utilized to solve one-dimensional wave equations in the formulation of the Maxwell's equations. Calculation procedure of the CIP method for three-dimensional scattering analysis is described in details. Update equations for boundary conditions of a perfectly conducting (PEC) interface and a dielectric interface are formulated and obtained in explicit forms. Numerical analyses of electromagnetic scatterings of PEC sphere, dielectric sphere and PEC cube are performed and the scattering coefficient is calculated and compared with the Mie's analytic results. As a result, the scattering coefficients show good agreement with the Mie's results, which demonstrates the validity of the CIP method and the formulated update equations. It is also shown that the phase of the scattering coefficients determined by the CIP method are slightly more accurate than that of the FDTD method.

Multiscale Computations for 3D Time-Dependent Maxwell's Equations in Composite Materials

We discuss multiscale computations of the initial-boundary value problem for three-dimensional time-dependent Maxwell's equations in composite materials. A multiscale algorithm is developed, and the convergence results are presented. Numerical simulations are carried out to validate the theoretical results reported in the present paper.

A CELL-VERTEX FINITE VOLUME TIME DOMAIN METHOD FOR ELECTROMAGNETIC SCATTERING

A cell-vertex based flnite volume scheme is used to solve the time-dependent Maxwell's equations and predict electromagnetic scattering from perfectly conducting bodies. The scheme is based on the cell-vertex flnite volume integration method, originally proposed by Ni (1), for solution of the two dimensional unsteady Euler equations of gas dynamics. The resulting solution is second-order accurate in space and time, and requires cell based ∞uctuations to be appropriately distributed to the state vector stored at cell vertices at each time step. Results are presented for two-dimensional canonical shapes and complex three dimensional geometries. Unlike in gas dynamics, no user deflned numerical damping is required in this novel cell-vertex based flnite volume integration scheme when applied to the time-domain Maxwell's equations.

The Landau–Lifshitz–Maxwell equation in dimension three

In dimension three, we show the existence of weak solutions (u, H, E) to the Landau-Lifshitz equation coupled with the time-dependent Maxwell equation such that u is Holder continuous away from a closed set Σ that has locally finite 3-dimensional parabolic Hausdorff measure. For two reduced Maxwell equations, Holder continuity of ∇u away from Σ is also established.

A High-Precision Integration Scheme for the Spectral-Element Time-Domain Method in Electromagnetic Simulation

A high-precision integration (HPI) scheme combined with the spectral-element time-domain (SETD) method is presented to solve the time-dependent Maxwell's equations. Spatial discretization by spectral elements will lead to block diagonal mass matrices, thus greatly alleviating the computational burden of inverting the mass matrices. A high-precision time integration method based on the matrix exponential is then employed to solve the discretized SETD system. Numerical examples demonstrate that this algorithm is unconditionally stable and very accurate.

Joly–Mercier boundary condition for the finite element solution of 3D Maxwell equations

Solving the time-dependent Maxwell equations in an unbounded domain requires the introduction of artificial absorbing boundary conditions ( ABCs) designed to minimize the amplitude of the parasitic waves reflected by the artificial frontier of the domain of computation. The construction of such ABCs needs to perform a rigorous mathematical and numerical analysis, in order to obtain a well-posed problem, from a mathematical point of view, and a stable algorithm, from a numerical point of view. In a previous study, Joly and Mercier (1989) [8] have proposed a new second-order ABC for Maxwell’s equation in dimension 3, well adapted to a variational approach. In this paper, we present how to apply the second-order ABC proposed in [8] in the framework of a finite element method.

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.

Stability analysis of FDTD to UPML for time dependent Maxwell equations

We study an finite-difference time-domain (FDTD) system of uniaxial perfectly matched layer (UPML) method for electromagnetic scattering problems. Particularly we analyze the discrete initial-boundary value problems of the transverse magnetic mode (TM) to Maxwell’s equations with Yee’s algorithm. An exterior domain in two spacial dimension is truncated by a square with a perfectly matched layer filled by a certain artificial medium. Besides, an artificial boundary condition is imposed on the outer boundary of the UPML. Using energy method, we obtain the stability of this FDTD system on the truncated domain. Numerical experiments are designed to approve the theoretical analysis.

A THREE-DIMENSIONAL TIME DOMAIN ELECTROMAGNETIC PARTICLE-IN-CELL CODE ON UNSTRUCTURED GRIDS

AbstractAccurate charged particle beam simulations are desired for modelling plasma physics problems or for designing microwave/accelarator technology devices. In complicated geometries, it might be preferable to use unstructured grids, for example, to be able to select between several issues in the design of high-power microwave devices. In this framework, we have proposed a numerical method for solving the three-dimensional time-dependent Vlasov—Maxwell equations on unstructured grids, based on a continuous Galerkin approximation. In this paper, we present numerical results obtained on characteristic three-dimensional applications: an open electromagnetic problem without any particle coupling and a microwave device example.

Implicit high-order unconditionally stable complex envelope algorithm for solving the time-dependent Maxwell's equations

Based on the Padé approximation and multistep method, we propose an implicit high-order unconditionally stable complex envelope algorithm to solve the time-dependent Maxwell's equations. Unconditional numerical stability can be achieved simultaneously with a high-order accuracy in time. As we adopt the complex envelope Maxwell's equations, numerical dispersion and dissipation are very small even at comparatively large time steps. To verify the capability of our algorithm, we compare the results of the proposed method with the exact solutions.

Application of operator splitting to the Maxwell equations including a source term

Motivated by numerical solution of the time-dependent Maxwell equations, we consider splitting methods for a linear system of differential equations w ′ ( t ) = A w ( t ) + f ( t ) , A ∈ R n × n split into two subproblems w 1 ′ ( t ) = A 1 w 1 ( t ) + f 1 ( t ) and w 2 ′ ( t ) = A 2 w 2 ( t ) + f 2 ( t ) , A = A 1 + A 2 , f = f 1 + f 2 . First, expressions for the leading term of the local error are derived for the Strang–Marchuk and the symmetrically weighted sequential splitting methods. The analysis, done in assumption that the subproblems are solved exactly, confirms the expected second order global accuracy of both schemes. Second, several relevant numerical tests are performed for the Maxwell equations discretized in space either by finite differences or by finite elements. An interesting case is the splitting into the subproblems w 1 ′ = A w 1 and w 2 ′ = f (with the split-off source term f). For the central finite difference staggered discretization, we consider second order splitting schemes and compare them to the classical Yee scheme on a test problem with loss and source terms. For the vector Nédélec finite element discretizations, we test the Gautschi–Krylov time integration scheme. Applied in combination with the split-off source term, it leads to splitting schemes that are exact per split step. Thus, the time integration error of the schemes consists solely of the splitting error.

Generalized Finite-Difference Time-Domain Method Utilizing Auxiliary Differential Equations for the Full-Vectorial Analysis of Photonic Crystal Fibers

We present the generalized finite-difference time-domain full-vectorial method by reformulating the time-dependent Maxwell's curl equations with electric flux density and magnetic field intensity, with auxiliary differential equations using complex-conjugate pole-residue pairs. The model is generic and robust to treat general frequency-dependent material and nonlinear material. The Sellmeier equation is implicitly incorporated as a special case of the general formulation to account for material dispersion of fused silica. The results are in good agreement with the results from the multipole method. Kerr nonlinearity is also incorporated in the model and demonstrated. Nonlinear solutions are provided for a one ring photonic crystal fiber as an example.

Interior penalty discontinuous Galerkin method for Maxwell's equations: optimal L2-norm error estimates

We consider the symmetric, interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell's equations in second-order form. In Grote et al. (2007, J. Comput. Appl. Math., 204, 375-386), optimal a priori estimates in the DG energy norm were derived, either for smooth solutions on arbitrary meshes or for low-regularity (singular) solutions on conforming, affine meshes. Here, we show that the DG methods are also optimally convergent in the L 2 -norm, on tetrahedral meshes and for smooth material coefficients. The theoretical convergence rates are validated by a series of numerical experiments in two-space dimensions, which also illustrate the usefulness of the interior penalty DG method for time-dependent computational electromagnetics.

Modeling of the electrode microwave discharge in nitrogen

Self-consistent two-dimensional modeling of a steady microwave discharge initiated at the end of the central electrode in nitrogen is presented. The discharge parameters are calculated at a gas pressure of 1 Torr and incident power of 30 W. The computational model includes the time-dependent Maxwell's equations, the balance equations describing the kinetics of charged and neutral plasma particles and the time-independent homogeneous Boltzmann equation for electrons. The processes involving vibrationally excited ground state molecular nitrogen are taken into account by the well-known analytic expression for the vibrational distribution of molecules in the diffusion approximation. It is shown that the spatially non-uniform microwave field causes the difference in plasma particles kinetics in different parts of the discharge. Results of numerical simulations for space distribution of electronically excited molecules have been found in good qualitative agreement with those taken from spectral measurements of first and second positive systems of nitrogen. Results confirm the concept according to which such a discharge comprises a self-sustained and a non-self-sustained discharge.

Quaternion analysis for generalized electromagnetic fields of dyons in an isotropic medium

Quaternion analysis of time-dependent Maxwell's equations in the presence of electric and magnetic charges has been developed and the solutions for the classical problem of moving charges (electric and magnetic) are obtained in a unique, simple and consistent manner.