Energy Stable Nodal DG Methods for Maxwell’s Equations of Mixed-Order Form in Nonlinear Optical Media

In this work,we develop energy stable numerical methods to simulate electromagnetic waves propagating in optical media where the media responses include the linear Lorentz dispersion,the instantaneous nonlinear cubic Kerr response,and the nonlinear delayed Raman molecular vibrational response.Unlike the first-order PDE-ODE governing equations considered previously in Bokil et al.(J Comput Phys 350:420–452,2017)and Lyu et al.(J Sci Comput 89:1–42,2021),a model of mixed-order form is adopted here that consists of the first-order PDE part for Maxwell’s equations coupled with the second-order ODE part(i.e.,the auxiliary differential equations)modeling the linear and nonlinear dispersion in the material.The main contribution is a new numerical strategy to treat the Kerr and Raman nonlinearities to achieve provable energy stability property within a second-order temporal discretization.A nodal discontinuous Galerkin(DG)method is further applied in space for efficiently handling nonlinear terms at the algebraic level,while preserving the energy stability and achieving high-order accuracy.Indeed with d_(E)as the number of the components of the electric field,only a d_(E)×d_(E)nonlinear algebraic system needs to be solved at each interpolation node,and more importantly,all these small nonlinear systems are completely decoupled over one time step,rendering very high parallel efficiency.We evaluate the proposed schemes by comparing them with the methods in Bokil et al.(2017)and Lyu et al.(2021)(implemented in nodal form)regarding the accuracy,computational efficiency,and energy stability,by a parallel scalability study,and also through the simulations of the soliton-like wave propagation in one dimension,as well as the spatial-soliton propagation and two-beam interactions modeled by the two-dimensional transverse electric(TE)mode of the equations.

The Maxwell-Heaviside Equations Explained by the Theory of Informatons

In the articles “Newtons Law of Universal Gravitation Explained by the Theory of Informatons” and “The Gravitational Interaction between Moving Mass Particles Explained by the Theory of Informatons” the gravitational interaction has been explained by the hypothesis that information carried by informatons is the substance of gravitational fields, i.e. the medium that the interaction in question makes possible. From the idea that “information carried by informatons” is its substance, it has been deduced that—on the macroscopic level—a gravitational field manifests itself as a dual entity, always having a field- and an induction component (Egand Bg) simultaneously created by their common sources. In this article we will mathematically deduce the Maxwell-Heaviside equations from the kinematics of the informatons. These relations describe on the macroscopic level how a gravitational field (Eg, Bg) is generated by whether or not moving masses and how spatial and temporal changes of Egand Bgare related. We show that there is no causal link between Egand Bg.

Analysis of Modeling the Influence of Electromagnetic Fields Radiated by Industrial Static Converters and Impacts on Operators Using Maxwell’s Equations

The study of Electromagnetic Compatibility is essential to ensure the harmonious operation of electronic equipment in a shared environment. The basic principles of Electromagnetic Compatibility focus on the ability of devices to withstand electromagnetic disturbances and not produce disturbances that could affect other systems. Imperceptible in most work situations, electromagnetic fields can, beyond certain thresholds, have effects on human health. The objective of the present article is focused on the modeling analysis of the influence of geometric parameters of industrial static converters radiated electromagnetic fields using Maxwell’s equations. To do this we used the analytical formalism for calculating the electromagnetic field emitted by a filiform conductor, to model the electromagnetic radiation of this device in the spatio-temporal domain. The interactions of electromagnetic waves with human bodies are complex and depend on several factors linked to the characteristics of the incident wave. To model these interactions, we implemented the physical laws of electromagnetic wave propagation based on Maxwell’s and bio-heat equations to obtain consistent results. These obtained models allowed us to evaluate the spatial profile of induced current and temperature of biological tissue during exposure to electromagnetic waves generated by this system. The simulation 2D results obtained from computer tools show that the temperature variation and current induced by the electromagnetic field can have a very significant influence on the life of biological tissue. The paper provides a comprehensive analysis using advanced mathematical models to evaluate the influence of electromagnetic fields. The findings have direct implications for workplace safety, potentially influencing standards and regulations concerning electromagnetic exposure in industrial settings.

Maxwell Equations and Magnetic Monopoles

The manuscript introduces an “ab initio” quantum model to deduce the Maxwell equations. After general considerations and laying out the model’s theoretical framework, these equations can be derived alongside a broad variety of other results. Specifically, a corollary of the present model proposes a possible mechanism underlying the formation of magnetic monopoles and allows estimating their formation energy in order of magnitude.

High-Order Spatial FDTD Solver of Maxwell’s Equations for Terahertz Radiation Production

We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma.

Complex Maxwell’s Equations

Maxwell’s equations in electromagnetism can be categorized into three dis-tinct groups based on the electromagnetic source when employing quaterni-ons. Each group represents a self-contained system in which Maxwell’s equations are applied and validated concurrently, in contrast to the previous approach that did not account for this. It has been noted that the formulation of these Maxwell equations ultimately results in the formulation of Max-well’s equations utilizing the scalar function.

基于Maxwell方程和R-L分数阶导数的高频激励条件下纳米晶材料磁滞特性预测模型

准确预测高频激励条件下纳米晶材料的磁滞特性,对电力电子变压器等高频电力设备磁性元件的优化设计具有重要意义。该文基于Maxwell方程和Riemann-Liouville(R-L)分数阶导数,建立高频激励下纳米晶材料磁滞特性预测模型。首先,根据Maxwell方程,首次推导出计及趋肤效应影响的涡流磁场强度解析式;其次,针对现有静态磁滞特性和异常效应建模方法未考虑趋肤效应影响的问题,依据R-L型分数阶导数理论,建立有效描述带材内部磁通密度分布不均时纳米晶静态磁滞特性与异常效应综合预测模型,并提出一种考虑频率效应的阻尼系数ρ和非整数阶n的解析计算方法;最后,基于磁场分离理论,建立高频激励下纳米晶材料磁滞特性预测模型。通过与纳米晶材料1K107B在高频激励下实测磁滞回线进行对比,验证该文所建预测模型的有效性。相较于经典模型,该文所建模型对磁滞回线的预测精度提升了35.8%。

STABILITY AND SUPER CONVERGENCE ANALYSIS OF ADI-FDTD FOR THE 2D MAXWELL EQUATIONS IN A LOSSY MEDIUM

Several new energy identities of the two dimenslonal（2D） Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved. These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction im- plicit finite difference time domain method for the 2D Maxwell equations （2D-ADI-FDTD）. It is proved that 2D-ADI-FDTD is approximately energy conserved, unconditionally sta- ble and second order convergent in the discrete L2 and H1 norms, which implies that 2D-ADI-FDTD is super convergent. By this super convergence, it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate. It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H1 norm. Experimental results to confirm the theoretical analysis on stability, convergence and energy conservation are presented.

Noether conserved quantities and Lie point symmetries of difference Lagrange-Maxwell equations and lattices

This paper presents a method to find Noether-type conserved quantities and Lie point symmetries for discrete mechanico-electrical dynamical systems,which leave invuriant the set of solutions of the corresponding difference scheme. This approach makes it possible to devise techniques for solving the Lagrange Maxwell equations in differences which correspond to mechanico-electrical systems,by adapting existing differential equations.In particular,it obtains a new systematic method to determine both the one-parameter Lie groups and the discrete Noether conserved quantities of Lie point symmetries for mechanico-electrical systems.As an application,it obtains the Lie point symmetries and the conserved quantities for the difference equation of a model that represents a capacitor microphone.

ANALYSIS OF FDTD TO UPML FOR MAXWELL EQUATIONS IN POLAR COORDINATES

An FDTD system associated with uniaxial perfectly matched layer（UPML） for an electromagnetic scattering problem in two-dimensional space in polar coordinates is considered.Particularly the FDTD system of an initial-boundary value problems of the transverse magnetic（TM） mode to Maxwell＇s equations is obtained by Yee＇s algorithm,and the open domain of the scattering problem is truncated by a circle with a UPML.Besides,an artificial boundary condition is imposed on the outer boundary of the UPML.Afterwards,stability of the FDTD system on the truncated domain is established through energy estimates by the Gronwall inequality.Numerical experiments are designed to approve the theoretical analysis.

Existence and Uniqueness of Global Smooth Solutions for Vlasov Maxwell Equations

Global existence of classical solutions to the relativistic Vlasov-Maxwell system, given sufficiently regular initial data, is a long-standing open problem. The aim of this project is to present in details the results of a paper published in 1986 by Robert Glassey and Walter Strauss. In that paper, a sufficient condition for the global existence of a smooth solution to the relativistic Vlasov-Maxwell system is derived. In the following, the resulting theorem is proved by taking initial data , . A small data global existence result is presented as well.

A Nonconforming Arbitrary Quadrilateral Finite Element Method for Approximating Maxwell's Equations

The main aim of this paper is to provide convergence analysis of Quasi-Wilson nonconforming finite element to Maxwell's equations under arbitrary quadrilateral meshes.The error estimates are derived,which are the same as those for conforming elements under conventional regular meshes.

A New Formulation of Maxwell’s Equations in Clifford Algebra

A new unification of the Maxwell equations is given in the domain of Clifford algebras with in a fashion similar to those obtained with Pauli and Dirac algebras. It is shown that the new electromagnetic field multivector can be obtained from a potential function that is closely related to the scalar and the vector potentials of classical electromagnetics. Additionally it is shown that the gauge transformations of the new multivector and its potential function and the Lagrangian density of the electromagnetic field are in agreement with the transformation rules of the second-rank antisymmetric electromagnetic field tensor, in contrast to the results obtained by applying other versions of Clifford algebras.

Adaptive Maxwell’s Equations Derived Optimization and Its Application in Antenna Array Synthesis

In this paper,a self-adaptive method for the Maxwell’s Equations Derived Optimization(MEDO)is proposed.It is implemented by applying the Sequential Model-Based Optimization(SMBO)algorithm to the iterations of the MEDO,and achieves the automatic adjustment of the parameters.The proposed method is named as adaptive Maxwell’s equations derived optimization(AMEDO).In order to evaluate the performance of AMEDO,eight benchmarks are used and the results are compared with the original MEDO method.The results show that AMEDO can greatly reduce the workload of manual adjustment of parameters,and at the same time can keep the accuracy and stability.Moreover,the convergence of the optimization can be accelerated due to the dynamical adjustment of the parameters.In the end,the proposed AMEDO is applied to the side lobe level suppression and array failure correction of a linear antenna array,and shows great potential in antenna array synthesis.

Analysis of Maxwell’s Equations. Cosmic Magnetism

According to Hypersphere World-Universe Model, dark matter particles DIRACs are magnetic dipoles consisting of two Dirac’s monopoles. We conclude that DIRACs are the subject of Maxwell’s equations. So-called “auxiliary” magnetic field intensity H is indeed current density of magnetic dipoles. The developed approach to magnetic field can explain a wealth of discovered phenomena in Cosmic Magnetism: a dark magnetic field, the large-scale structure of the Milky Way’s magnetic field, and other magnetic phenomena which are only partly related to objects visible in other spectral ranges.

New Reflection Principles for Maxwell's Equations and Their Applications

Some new reflection principles for Maxwell's equations are first established, which are then applied to derive two novel identifiability results in inverse electromagnetic obstacle scattering problems with polyhedral scatterers.

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 un-derlying 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 dis-cretization 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.

Complex Maxwell's equations

A unified complex model of Maxwell＇s equations is presented.The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities.A new vacuum solution is obtained,and a new transformation under which Maxwell＇s equations are invariant is proposed.This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential.An electric dipole moment is found to be related to the magnetic charges,and Dirac＇s quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges.We generalize Maxwell＇s equations to include longitudinal waves.A formal analogy between this formulation and Dirac＇s equation is also discussed.

Nonlinear waves of the Hirota and the Maxwell Bloch equations in nonlinear optics

In this paper, considering the Hirota and the Maxwell–Bloch （H-MB） equations which are governed by femtosecond pulse propagation through a two-level doped fiber system, we construct the Darboux transformation of this system through a linear eigenvalue problem. Using this Daurboux transformation, we generate multi-soliton, positon, and breather solutions （both bright and dark breathers） of the H-MB equations. Finally, we also construct the rogue wave solutions of the above system.

A wavelet multiscale method for inversion of Maxwell equations

This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.