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.     

Solving metamaterial Maxwell’s equations via a vector wave integro-differential equation

In this paper, we discuss the time-domain metamaterial Maxwell’s equations. One major contribution of this paper is that after some effort we find that the metamaterial Maxwell’s equations can be beautifully reduced to a vector wave integro-differential equation involving just one unknown, which is quite similar to that obtained from the standard Maxwell’s equations in vacuum. Then we study the existence and uniqueness of this new modeling equations, and propose a fully-discrete finite element method to solve this model. Numerical results justifying our analysis are presented. This discovery shall make simulation of metamaterials much more efficient than the previous works.     

An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics

In plasma simulations, where the speed of light divided by a characteristic length is at a much higher frequency than other relevant parameters in the underlying system, such as the plasma frequency, implicit methods begin to play an important role in generating efficient solutions in these multi-scale problems. Under conditions of scale separation, one can rescale Maxwell’s equations in such a way as to give a magneto static limit known as the Darwin approximation of electromagnetics. In this work, we present a new approach to solve Maxwell’s equations based on a Method of Lines Transpose (\(\hbox {MOL}^T\)) formulation, combined with a fast summation method with computational complexity \(O(N\log {N})\), where N is the number of grid points (particles). Under appropriate scaling, we show that the proposed schemes result in asymptotic preserving methods that can recover the Darwin limit of electrodynamics.     

An Adaptive P 1 Finite Element Method for Two-Dimensional Maxwell’s Equations

Recently a new numerical approach for two-dimensional Maxwell’s equations based on the Hodge decomposition for divergence-free vector fields was introduced by Brenner et al. In this paper we present an adaptive P ₁ finite element method for two-dimensional Maxwell’s equations that is based on this new approach. The reliability and efficiency of a posteriori error estimators based on the residual and the dual weighted-residual are verified numerically. The performance of the new approach is shown to be competitive with the lowest order edge element of Nédélec’s first family.     

Space–time mesh refinement for elastodynamics. Numerical results

We propose here a generalization to the elastodynamic equations of the space–time mesh refinement technique introduced in [Raffinement de maillage spatio-temporel pour les équations de Maxwell, Ph.D. thesis, Université de Dauphine, Paris, 2000] for the Maxwell’s equations. This method uses a discrete energy conservation to ensure the stability. The method is presented in a variational way applicable to other type of hyperbolic systems. Several numerical experiments are provided to show the efficiency of this approach.     

Implicit-explicit time integration of a high-order particle-in-cell method with hyperbolic divergence cleaning

A high-order implicit-explicit additive Rung-Kutta time integrator is implemented in a particle-in-cell method based on a high-order discontinuous Galerkin Maxwell solver for the simulation of plasmas. The method enforces Gauss law by a hyperbolic divergence cleaner that transports divergence out of the computational domain at several times the speed of light. The stiffness in the field equations induced by this high transport speeds is alleviated by an implicit time integration, while an explicit time integration ensures a computationally efficient particle update. Simulations on a plasma wave and a Weibel instability show that the implicit-explicit solver is computationally efficient, allowing for computations with high divergence transport speeds that ensure an accurate representation of the governing plasma equations. The high-order method only requires two time steps per plasma wave period. Numerical instability appears when the time step exceeds the plasma frequency time scale. A divergence transport speed of approximately ten times the speed of light is shown to be optimal, since it combines an accurate representation of Gauss law with a small influence of numerical noise on the solution.     

Solution of time-domain Maxwell equation with PML by using modified Laguerre polynomials

In this work, a stable numerical algorithm proposed by Chung et al. for the time-domain Maxwell equations is generalized. The time-domain Maxwell equations are solved by expressing the transient behaviors in terms of the modified Laguerre polynomials, and then the original equations of the initial value and boundary value can be transformed into a series of problems independent of the time variable. In this case the method of finite difference （FD）, the finite element method （FEM）, the method of moment （MoM）, etc. or the combination of these methods can be used to solve the problems. Finally, a numerical model is provided for the scattering problem with perfect matched layer （PML） by using FD. The comparison between the results of the proposed method and FDTD is presented to verify the proposed new method.     

Approximating thermo-viscoelastic heating of largely strained solid rubber components

Mechanically induced viscoelastic dissipation is difficult to compute when the constitutive model is defined by history integrals. The computation of the viscous energy dissipated is in the form of a double convolution integral. In this study, we present a method to approximate the dissipation for constitutive models in history integral form that represent Maxwell-like materials. The dissipation is obtained without directly computing the double convolution integral. The approximation requires that the total stress can be separated into elastic and viscous components, and that the relaxation form of the constitutive law is defined with a Prony series. A numerical approach often taken to approximate a history integral involves interpolating the history integral’s kernel across a time step. Integration then yields finite difference equations for the evolution of the viscous stresses in time. In the case when the material is modeled with a Prony series, the form of these finite difference equations is similar to the form of the finite difference equations for a Maxwell solid. Since the dissipation rate in a Maxwell solid can be easily computed from knowledge of its viscous stress and the Prony series constants (spring-dashpot constants), we computationally investigated employing a Maxwell solid’s dissipation function to couple thermal and large strain history integral based finite element models of solid rubber components. Numerical data is provided to support this analogy and to help understand its limitations. A rubber cylinder with an imbedded steel disk is dynamically loaded, and the non-uniform heating within the cylinder is computed.     

Numerical modelling of pulsar magnetospheres

Numerical models used in the study of the pulsar magnetosphere are described: a vacuum model based only on Maxwell's equations and a more realistic model employing both Maxwell's and relativistic two-fluid equations. The general approach to solving the chosen sets of partial differential equations is outlined and the possible boundary conditions are examined. Numerical methods suitable for solving Maxwell's equations are discussed and a method is developed for solving the combined fluid plus Maxwell model. Results are presented and discussed and the possible improvements in the approach are indicated.     

MNPBEM – A Matlab toolbox for the simulation of plasmonic nanoparticles

MNPBEM is a Matlab toolbox for the simulation of metallic nanoparticles (MNP), using a boundary element method (BEM) approach. The main purpose of the toolbox is to solve Maxwell?s equations for a dielectric environment where bodies with homogeneous and isotropic dielectric functions are separated by abrupt interfaces. Although the approach is in principle suited for arbitrary body sizes and photon energies, it is tested (and probably works best) for metallic nanoparticles with sizes ranging from a few to a few hundreds of nanometers, and for frequencies in the optical and near-infrared regime. The toolbox has been implemented with Matlab classes. These classes can be easily combined, which has the advantage that one can adapt the simulation programs flexibly for various applications.Program summaryProgram title: MNPBEMCatalogue identifier: AEKJ_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEKJ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen?s University, Belfast, N. IrelandLicensing provisions: GNU General Public License v2No. of lines in distributed program, including test data, etc.: 15 700No. of bytes in distributed program, including test data, etc.: 891 417Distribution format: tar.gzProgramming language: Matlab 7.11.0 (R2010b)Computer: Any which supports Matlab 7.11.0 (R2010b)Operating system: Any which supports Matlab 7.11.0 (R2010b)RAM: ?1 GByteClassification: 18Nature of problem: Solve Maxwell?s equations for dielectric particles with homogeneous dielectric functions separated by abrupt interfaces.Solution method: Boundary element method using electromagnetic potentials.Running time: Depending on surface discretization between seconds and hours.     

Field equations for incompressible non-viscous fluids using artificial intelligence

In this article, we introduce new field equations for incompressible non-viscous fluids, which can be treated similarly to Maxwell’s electromagnetic equations based on artificial intelligence algorithms. Lagrangian and Hamiltonian formulations are used to arrive at field equations that are solved using convolutional neural networks. Four linear differential equations, which describe the two fields, namely, the dynamic pressure and the vortex fields, are derived, and these can be used in place of Euler’s equation. The only assumption while deriving this equation is that the dynamic pressure and vortex fields obey the superposition principle. The important finding to be noted is that Euler’s fluid equations can be converted into field equations analogous to Maxwell’s electromagnetic equations. We solve the flow problem for laminar flow past a cylinder, sphere, and cone in two dimensions similar to the conduction in a uniform electric field and arrive at closed-form expressions. These closed-form expressions, which are obtained for the potentials of fluid flow, are similar to the streamline potential functions in the case of fluid dynamics.

     

Unified matrix-exponential FDTD formulations for modeling electrically and magnetically dispersive materials

Unified matrix-exponential finite difference time domain (ME-FDTD) formulations are presented for modeling linear multi-term electrically and magnetically dispersive materials. In the proposed formulations, Maxwell?s curl equations and the related dispersive constitutive relations are cast into a set of first-order differential matrix system and the field?s update equations can be extracted directly from the matrix-exponential approximation. The formulations have the advantage of simplicity as it allows modeling different linear dispersive materials in a systematic manner and also can be easily incorporated with the perfectly matched layer (PML) absorbing boundary conditions (ABCs) to model open region problems. Apart from its simplicity, it has been shown that the proposed formulations necessitate less storage requirements as compared with the well-know auxiliary differential equation FDTD (ADE-FDTD) scheme while maintaining the same accuracy performance.     

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations. We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.     

Three-Dimensional Numerical Model of the Interaction of Laser Radiation with the Plasma of a Plastic Target

The effect of generation of high-energy protons and carbon nuclei is identified in the threedimensional model of the interaction of a high-power electromagnetic field with plasma with supercritical density by the numerical solution of the Vlasov–Maxwell equations. The effect was first discovered in 2000 in experiments carried out at the Lawrence Livermore National Laboratory (United States) using the petawatt laser.     

Using two types of computer algebra systems to solve maxwell optics problems

To synthesize Maxwell optics systems, the mathematical apparatus of tensor and vector analysis is generally employed. This mathematical apparatus implies executing a great number of simple stereotyped operations, which are adequately supported by computer algebra systems. In this paper, we distinguish between two stages of working with a mathematical model: model development and model usage. Each of these stages implies its own computer algebra system. As a model problem, we consider the problem of geometrization of Maxwell’s equations. Two computer algebra systems—Cadabra and FORM—are selected for use at different stages of investigation.     

A lattice Boltzmann model for electromagnetic waves propagating in a one-dimensional dispersive medium

A first-order extended lattice Boltzmann (LB) model with special forcing terms for one-dimensional Maxwell equations exerting on a dispersive medium, described either by the Debye or Drude model, is proposed in this study. The time dependent dispersive effect is obtained by the inverse Fourier transform of the frequency-domain permittivity and is incorporated into the LB evolution equations via equivalent forcing effects. The Chapman–Enskog multi-scale analysis is employed to ensure that proposed scheme is mathematically consistent with the targeted Maxwell’s equations at the macroscopic limit. Numerical validations are executed through simulating four representative cases to obtain their LB solutions and compare those with the analytical solutions and existing numerical solutions by finite difference time domain (FDTD). All comparisons show that the differences in numerical values are very small. The present model can thus accurately predict the dispersive effects, and demonstrate first order convergence. In addition to its accuracy, the proposed LB model is also easy to implement. Consequently, this new LB scheme is an effective approach for numerical modeling of EM waves in dispersive media.     

Simulating Plasma Microwave Diagnostics

Computational simulation of plasma diagnostics via microwave absorption has been successfully accomplished. This simulation capability is developed from solutions to a combination of the three-dimensional Maxwell equations and the generalized Ohm’s law in the time domain. As the simulation procedure developed, numerical results were obtained for a range of plasma transport properties including electrical conductivity, permittivity, and plasma frequency. The present results reveal the wave reflection at the media interface and substantial distortion of the electromagnetic field within a thin plasma sheet from a guided microwave. The present numerical simulation also accurately predicts the microwave blackout phenomenon as the wave propagates through a thick plasma sheet. The diffractions and refractions occurring at antenna apertures and passing through a plasma column are captured numerically. Finally, the numerical simulation has successfully duplicated a plasma diagnostic experiment in a hypersonic magneto-hydrodynamic channel.     

Discontinuous Hamiltonian Finite Element Method for Linear Hyperbolic Systems

We develop a Hamiltonian discontinuous finite element discretization of a generalized Hamiltonian system for linear hyperbolic systems, which include the rotating shallow water equations, the acoustic and Maxwell equations. These equations have a Hamiltonian structure with a bilinear Poisson bracket, and as a consequence the phase-space structure, “mass” and energy are preserved. We discretize the bilinear Poisson bracket in each element with discontinuous elements and introduce numerical fluxes via integration by parts while preserving the skew-symmetry of the bracket. This automatically results in a mass and energy conservative discretization. When combined with a symplectic time integration method, energy is approximately conserved and shows no drift. For comparison, the discontinuous Galerkin method for this problem is also used. A variety numerical examples is shown to illustrate the accuracy and capability of the new method.     

Iterative addition of parallel temperature effects to finite-difference simulation of radio-frequency wave propagation in plasmas

Accurate simulations of how radio frequency (RF) power is launched, propagates, and absorbed in a magnetically confined plasma is a computationally challenging problem that for which no comprehensive approach presently exists. The underlying physics is governed by the Vlasov–Maxwell equations, and characteristic length scales can vary by three orders of magnitude. Present algorithms are, in general, based on finding the constituative relation between the induced RF current and the RF electric field and solving the resulting set of Maxwell’s equations. These linear equations use a Fourier basis set that is not amenable to multi-scale formulations and have a large dense coefficient matrix that requires a high-communications overhead factorization technique. Here the use of operator splitting to separate the current and field calculations, and a low-overhead iterative solver leads to an algorithm that avoids these issues and has the potential to solve presently intractable problems due to its data-parallel and favorable scaling characteristics. We verify the algorithm for the iterative addition of parallel temperature effects for a 1D electron Langmuir by reproducing the solution obtained with the existing Fourier kinetic RF code aorsa (Jaeger et al., 2008).     

Invariant solutions of integro-differential Vlasov–Maxwell equations in Lagrangian variables by Lie group analysis

The Lie point symmetries of the Vlasov–Maxwell system in Lagrangian variables are investigated by using a direct method for symmetry group analysis of integro-differential equations, with emphasis on solving nonlocal determining equations. All similarity reduction forms for the system are obtained by using different approaches and some analytical and numerical solutions are presented.