Maxwell System Research Articles

Electron-scale temperature gradients in kinetic equilibrium: MMS observations and Vlasov–Maxwell solutions

For an ideal gas, the notion of thermodynamic equilibrium requires that the system's temperature be spatially uniform. We demonstrate that for a collisionless plasma, kinetic equilibrium can be achieved even in the presence of a temperature gradient. Motivated by recent Magnetospheric Multiscale (MMS) magnetopause observations of thin current layers exhibiting electron-scale temperature gradients, here we present an exact solution to the Vlasov–Maxwell system that accounts for the observed electron temperature gradient and reproduces much of the plasma's spatial variation along one dimension. The current layer in the Vlasov–Maxwell model is self-consistently supported by a prominent crescent-shaped electron velocity distribution that naturally arises due to the temperature transition, which agrees with the electron distribution structures measured by MMS during two asymmetric current layers encountered at the magnetopause.

Higher-order accurate and conservative hybrid numerical scheme for multi-variables time-fractional Vlasov-Maxwell system: An Atangana-Baleanu Caputo approach

With the historical review, it can determine that plasma scientists devolved many numerical schemes in literature to solve the kinetic plasma models such as Vlasov- Maxwell system, but the fatal problem is that most of the designed algorithms violate the conservation laws. This issue is significant and cannot disregard because the Vlasov Maxwell system is connected with conservations properties and can evaluate after performing some effective mathematical operations. In this prospective, we formulated the Atangana-Baleanu fractional derivative with Caputo concepts based fractional-order plasma model to study the fractional performance of the plasma particles and further suggested a higher-order conservative numerical algorithm that follows the conservation laws, based on operational matrices theory which is evaluated by using the concepts of Shifted Gegenbauer estimations. The numerical convergence of the designed algorithm is also inspected to validate the competency and compatibility. The comprehensive analysis of the fractional performance of the plasma particles is also part of the study. Moreover, this concept can extend to the variable order and multi-dimensional problems for Vlasov and Boltzmann system.

Homogenization of a non-stationary periodic Maxwell system in the case of constant permeability

In L2(R3;C3), we consider a selfadjoint operator Lε, ε>0, given by the differential expression μ−1/2curlη(x/ε)−1curlμ−1/2−μ1/2∇ν(x/ε)divμ1/2, where μ is a constant positive matrix, a matrix-valued function η(x) and a real-valued function ν(x) are periodic with respect to some lattice, positive definite and bounded. We study the behavior of the operator-valued functions cos⁡(τLε1/2) and Lε−1/2sin⁡(τLε1/2) for τ∈R and small ε. It is shown that these operators converge to the corresponding operator-valued functions of the operator L0 in the norm of operators acting from the Sobolev space Hs (with a suitable s) to L2. Here L0 is the effective operator with constant coefficients. Also, an approximation with corrector in the (Hs→H1)-norm for the operator Lε−1/2sin⁡(τLε1/2) is obtained. We prove error estimates and study the sharpness of the results regarding the type of the operator norm and regarding the dependence of the estimates on τ. The results are applied to homogenization of the Cauchy problem for the non-stationary Maxwell system in the case where the magnetic permeability is equal to μ, and the dielectric permittivity is given by the matrix η(x/ε).

Novel tensorial Thixo-Visco-Plastic framework for rheological characterization of human blood

Characterizing human blood, a complex material with a spectrum of thixo-elasto-visco-plastic properties, through the development of more effective and efficient models has achieved special interest of late. This effort details the development a new approach, the tensorial-enhanced-Thixo-Visco-Plastic model (t-e-TVP), which integrates elements from the proven Bingham and generalized Maxwell systems to create a more robust framework and subsequently cast into a tensorial format. Here, the elastic and viscoelastic stress contributions from the microstructure are superimposed upon the viscoelastic backbone solution for stress offered by the modified TVP frame. The utility of this novel model is tested against the contemporary tensorial-ethixo-mHAWB (t-ethixo-mHAWB) framework, a similar model with a greater number of parameters, using rheological data of human blood collected on an ARESG2 strain-controlled rheometer. The blood samples are parametrically and statistically analyzed, entailing the comparison of the t-e-TVP and t-ethixo-mHAWB models with their capacity to accurately predict small and large amplitude oscillatory shear as well as unidirectional large amplitude oscillatory shear flow in blood.

Global Strichartz estimates for an inhomogeneous Maxwell system

We show global-in-time Strichartz estimates for the isotropic Maxwell system with divergence free data. On the scalar permittivity and permeability we impose decay assumptions as and a non-trapping condition. The proof is based on smoothing estimates in weighted L 2 spaces which follow from corresponding resolvent estimates for the underlying Helmholtz problem.

Higher-Dimensional Fractional Order Modelling for Plasma Particles with Partial Slip Boundaries: A Numerical Study.

We integrate fractional calculus and plasma modelling concepts with specific geometry in this article, and further formulate a higher dimensional time-fractional Vlasov Maxwell system. Additionally, we develop a quick, efficient, robust, and accurate numerical approach for temporal variables and filtered Gegenbauer polynomials based on finite difference and spectral approximations, respectively. To analyze the numerical findings, two types of boundary conditions are used: Dirichlet and partial slip. Particular methodology is used to demonstrate the proposed scheme’s numerical convergence. A detailed analysis of the proposed model with plotted figures is also included in the paper.

Regularity theory for nonautonomous Maxwell equations with perfectly conducting boundary conditions

In this work we study linear Maxwell equations with time- and space-dependent matrix-valued permittivity and permeability on domains with a perfectly conducting boundary. This leads to an initial boundary value problem for a first-order hyperbolic system with characteristic boundary. We prove a priori estimates for solutions in Hm. Moreover, we show the existence of a unique Hm-solution if the coefficients and the data are sufficiently regular and satisfy certain compatibility conditions. Since the boundary is characteristic for the Maxwell system, we have to exploit the divergence conditions in the Maxwell equations in order to derive the energy-type Hm-estimates. A combination of these estimates with several regularization techniques then yields the existence of solutions in Hm.

Sign-changing solutions for a kind of Klein–Gordon–Maxwell system

In this paper, the subcritical Klein–Gordon–Maxwell system −Δu + V(x)u − (2ω + ϕ)ϕu = f(u) coupled with Δϕ=(ω+ϕ)u2,x∈R3 is studied. Under general assumptions on V, f, by using the abstract critical theorems, which are derived from the minimax method and the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions for this system, in particular, a sequence of high energy sign-changing solutions is obtained if f is odd.

Asymptotics for 2D whispering gallery modes in optical micro-disks with radially varying index

Abstract Whispering gallery modes [WGM] are resonant modes displaying special features: they concentrate along the boundary of the optical cavity at high polar frequencies and they are associated with complex scattering resonances very close to the real axis. As a classical simplification of the full Maxwell system, we consider 2D Helmholtz equations governing transverse electric or magnetic modes. Even in this 2D framework, very few results provide asymptotic expansion of WGM resonances at high polar frequency $m\to \infty $ for cavities with radially varying optical index. In this work, using a direct Schrödinger analogy, we highlight three typical behaviors in such optical micro-disks, depending on the sign of an ‘effective curvature’ that takes into account the radius of the disk and the values of the optical index and its derivative. Accordingly, this corresponds to abruptly varying effective potentials (step linear or step harmonic) or more classical harmonic potentials, leading to three distinct asymptotic expansions for ground state energies. Using multiscale expansions, we design a unified procedure to construct families of quasi-resonances and associate quasi-modes that have the WGM structure and satisfy eigenequations modulo a super-algebraically small residual ${\mathscr{O}}(m^{-\infty })$. We show using the black box scattering approach that quasi-resonances are ${\mathscr{O}}(m^{-\infty })$ close to true resonances.

Reduced-Order Models for CAD: Shrinking Electromagnetics into a Simple Circuit

A new model order reduction strategy based on the reduced-basis method is carried out in this work. Starting off from time-harmonic Maxwell's equations, a new representation of the original Maxwell system is developed. First, a reduced basis approximation allows for a reduced-order representation of electrodynamics in the frequency band of interest. As a result, the Kurokawa series representation for electromagnetics turns pretty much into a finite sum of dominant eigenresonances, which stand upon global eigenmodes of the Maxwell system. This gives rise to a linear dynamical system in electromagnetics and, after a proper arrangement, provides extremely useful physical information from which an electrical engineer can get actionable design insights. In this work, we use computational electromagnetics as an actual design tool and several realistic design applications will be considered during the presentation.

Justification of the asymptotic coupled mode approximation of out-of-plane gap solitons in Maxwell equations

In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov–Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in Dohnal and Dörfler, [2013 Multiscale Model. Simul. 11 162–191].

Gauge-symmetrization method for energy-momentum tensors in high-order electromagnetic field theories

For electromagnetic field theories, canonical energy-momentum conservation laws can be derived from the underpinning spacetime translation symmetry according to the Noether procedure. However, the canonical Energy-Momentum Tensors (EMTs) are neither symmetric nor gauge-symmetric (gauge invariant). The Belinfante-Rosenfeld (BR) method is a well-known procedure to symmetrize the EMTs, which also renders them gauge symmetric for first-order field theories. High-order electromagnetic field theories appear in the study of gyrokinetic systems for magnetized plasmas and the Podolsky system for the radiation reaction of classical charged particles. For these high-order field theories, gauge-symmetric EMTs are not necessarily symmetric and vice versa. In the present study, we develop a new gauge-symmetrization method for EMTs in high-order electromagnetic field theories. The Noether procedure is carried out using the Faraday tensor F, instead of the 4-potential A, to derive a canonical EMT T_N. We show that the gauge-dependent part of T_N can be removed using the displacement-potential tensor \mathcal{F}=\mathcal{D}*A/4\pi, where \mathcal{D} is the anti-symmetric electric displacement tensor. This method gauge-symmetrize the EMT without necessarily making it symmetric, which is adequate for applications not involving general relativity. For first-order electromagnetic field theories, such as the standard Maxwell system, \mathcal{F} reduces to the familiar BR super-potential \mathcal{S}, and the method developed can be used as a simpler procedure to calculate \mathcal{S} without employing the angular momentum tensor in 4D spacetime. When the electromagnetic system is coupled to classical charged particles, the gauge-symmetrization method for EMTs is shown to be effective as well.

Limit model for the Vlasov–Maxwell system with strong magnetic fields via gyroaveraging

This paper deals with the Vlasov–Maxwell system in the case of a strong magnetic field. After a physically motivated nondimensionalization of the original system, a Hilbert expansion is employed around a small parameter given as the product of the characteristic time scale and the gyrofrequency. From this, necessary conditions on the solvability of the reduced system are derived. An important aspect is the reduction of the six-dimensional phase space to five dimensions. In addition to the discussion of the partial differential equations, also initial and boundary conditions both for the full system and the limit model are studied.

Radiation and scattering in electromagnetic waveguides near thresholds

A waveguide occupies a three-dimensional domain G G with several cylindrical outlets to infinity and is described by the stationary Maxwell system with perfectly conductive boundary conditions. It is assumed that the medium filling the waveguide is homogeneous and isotropic at infinity in a limiting sense. The paper is devoted to description of the behavior of the scattering matrix, radiation conditions, and solutions as the spectral parameter tends to a threshold. In particular, it is shown that the scattering matrix has finite one-sided limits at every threshold and the limits are expressed in terms of the “scattering matrix stable near the threshold”.

The coupled heat Maxwell equations with temperature-dependent permittivity

We consider the heat equation coupled with the Maxwell system, the Ampère-Maxwell equation being coupled to the heat equation by the permittivity, which depends on the temperature due to thermal agitation, and the heat equation being coupled to the Maxwell system by the volumic heat source term. Our purpose is to establish the existence of a local-in-time solution to this coupled problem. Firstly, fixing the temperature distribution, we study the resulting Maxwell system, a nonautonomous system due to the dependence of the permittivity on the temperature and consequently on time, by using the theory of evolution systems. Next, we return to our coupled problem, introducing a fixed-point problem in the closed convex set K(0;R):={z∈B¯(0;R);z(0)=0} of the Banach space C1([0,T];C1(Ω¯)) and proving that the hypotheses of Schauder's theorem are verified for R sufficiently large. The construction of the fixed-point problem is nontrivial as we need K(0;R) to be stable.

Explicit high-order exponential time integrator for discontinuous Galerkin solution of Maxwell's equations

In this paper, exponential propagator (EP) is introduced to the discontinuous Galerkin finite element method (DG-FEM) for solving transient electromagnetic problems. As a kind of explicit time-stepping method, the proposed EP has fourth order accuracy in time and requires only one storage unit for each of the degrees-of-freedom. It provides an attractive alternative to the high-order low-storage explicit Runge-Kutta (LSERK) approach, which is the most popular time-marching scheme currently used in high-fidelity simulation of Maxwell's equations with DG-FEM. When centered numerical flux is utilized, the discretized Maxwell's system in lossless region is symplectic structure-preserving (energy conserving), and thus very suitable for accurately computing the long-time propagation of waves in non-dissipative medium, or simulating the motion of charged particles in electromagnetic fields. Our method can also handle upwind numerical flux at the sacrifice of symplectic-conserving virtue for higher accuracy in spatial discretization and spurious inhibition. A major contribution of this work is that, the application range of this scheme is extended to lossy media such as conductors and particularly unsplit perfectly matched layers in the framework of DG-FEM. This greatly facilitates the treatment of open boundary conditions in scattering and/or radiation problems. Extensive numerical examples, including the analysis of photonic crystals, dielectric coated perfect conducting cylinder, propagation of waves in zero-index materials, and corrugated O-type Cherenkov oscillators etc., are presented to demonstrate its stability, accuracy, and performance. All the numerical experiments show that the presented method is superior to the LSERK scheme when centered numerical flux is adopted.

An alternative to the Simon tensor

The Simon tensor gives rise to a local characterization of the Kerr-NUT family in the stationary class of vacuum spacetimes. We find that a symmetric and traceless tensor in the quotient space of the stationary Killing trajectory offers a useful alternative to the Simon tensor. Our tensor is distinct from the spatial dual of the Simon tensor and illustrates the geometric property of the three dimensional quotient space more manifest. The reconstruction procedure of the metric for which the generalized Simon tensor vanishes is spelled out in detail. We give a four dimensional description of this tensor in terms of the Coulomb part of the imaginary selfdual Weyl tensor, which corresponds to the generalization of the three-index tensor defined by Mars. This allows us to establish a new and simple criterion for the Kerr-NUT family: the gradient of the Ernst potential becomes the non-null eigenvector of the Coulomb part of the imaginary selfdual Weyl tensor. We also discuss the SU(1, 2) covariant extension of the obstruction tensor into the Einstein–Maxwell system as an intrinsic characterization of the Kerr–Newman-NUT family.

Conservative semi-Lagrangian kinetic scheme coupled with implicit finite element field solver for multidimensional Vlasov Maxwell system

In this paper, a conservative semi-Lagrangian (CSL) kinetic scheme coupled with an implicit finite element method (IFEM) is developed for multidimensional electromagnetic plasma simulations. The present method (CSL-IFEM) enjoys the respective advantages of the CSL and IFEM, including mass conservation, high-order spatial accuracy, as well as being free from the Courant-Friedrichs-Lewy (CFL) condition. In present CSL-IFEM, the CSL scheme with a general high order positivity-preserving limiter is utilized for the spatial discretization of the Vlasov equation, which enables the proposed method to remove the CFL limitation in phase space, exactly conserve mass, and preserve the positivity of the distribution function. The IFEM solver is developed for the spatial discretization of the Maxwell equation, which makes the current method free from CFL limitation induced by the speed of light and easily handles general field boundary conditions. To make the proposed CSL-IFEM more efficient in multidimensional simulations, the dimensional splitting procedure and sparse matrix technology are implemented in CSL and IFEM, respectively. Then the Vlasov solver CSL and Maxwell solver IFEM are coupled via the current density calculated from the general Ohm law. Finally, several numerical experiments, including electromagnetic wave (2d0v), particle gyromotion (0d2v), streaming Weibel instability (1d2v) and magnetic reconnection (2d3v), are performed to demonstrate the capabilities of the proposed method.

Seismic responses of adjacent bridge structures coupled by tuned inerter damper

This paper proposes using a tuned inerter damper (TID) system to mitigate the potential pounding and unseating damages between the adjacent bridge structures under severe earthquakes. The control effectiveness of the proposed method is investigated both in the frequency and time domains. For comparison, the models and responses of the adjacent bridge structures without control and controlled by the traditional viscous dampers that are modelled by a Kelvin system or a Maxwell system are also developed and calculated. The analytical results reveal that the TID system can achieve almost the same or even better control effectiveness compared to the conventional viscous dampers with a much smaller additional damping. The proposed method can provide a good alternative to control the excessive relative motions between the adjacent bridge structures.

Comparison between fluid-gravity and membrane-gravity dualities for Einstein–Maxwell system

Derivative expansion and large-D expansion are two perturbation techniques, which are used to generate dynamical black-brane solutions to Einstein’s equations in presence of negative cosmological constant. In this paper we have compared these two techniques and established the equivalence of the gravity solutions generated by these two different techniques in appropriate regime of parameter space up to first non-trivial order in both the perturbation parameters for Einstein–Maxwell systems, generalizing the earlier works of [, ] for non-charged systems. An one-to-one map between dynamical black-brane geometry and AdS space, which also exists at finite number of dimensions, has also been established.