Time-dependent Maxwell Equations Research Articles

Nodal discontinuous Galerkin methods for Maxwell's equations in Lorentz-Kerr-Raman medium without nonlinear algebraic solver

The propagation of electromagnetic waves is modeled by time-dependent Maxwell's equations coupled with constitutive laws that describe the responses of the media. In this work, we consider a nonlinear model that describes the electromagnetic wave in an optical medium with the linear Lorentz effect and the cubic nonlinear instantaneous Kerr and delayed Raman effects. Mathematically this model obeys an energy conservative/dissipative law. Though there have been active efforts in designing numerical methods (e.g. of finite difference / finite element /discontinuous Galerkin type) to simulate this model, the methods proposed here are distinctive in that they are free of any nonlinear algebraic solvers. Moreover, in the absence of the Raman effect, our methods also enjoy a provable discrete energy law, and optimal a priori error estimates are further established when the exact solutions are sufficiently smooth. The key ingredients of the new methods include some novel treatment in time discretizations and nodal discontinuous Galerkin spatial discretization for the specific nonlinearities, and they also render a local nature of the methods and hence their suitability for parallel implementation with great efficiency. Numerical experiments are performed to illustrate the accuracy, stability, computational efficiency and parallel scalability of the proposed methods. We further apply the methods to simulate some physically relevant problems in one, two, and three dimensions.

Splines finite element solver for one-dimensional time-dependent Maxwell's equations via Fourier Transform Discretization

In this article, we solve the time-dependent Maxwell coupled equations in their one-dimensional version relatively to space-variable. We effectuate a variable reduction via Fourier transform to make the time variable as a frequency parameter easy and quickly to manage. A Galerkin variational method based on higher-order spline interpolations is used to approximate the solution relatively to the spacial variable. So, the state of existence of the solution, its uniqueness, and its regularity are studied and proved, and the study is also provided by an error estimate and the order of convergence of the proposed method. Also, we use the critical Nyquist frequency to calculate numerically the solution of the Inverse Fourier Transform(IFT); and for all numerical computations, we consider several quadrature methods. Finally, we give some experiments to illustrate the success of such an approach.

Fast parallel IGA-ADS solver for time-dependent Maxwell's equations

We propose a simulator for time-dependent Maxwell's equations with linear computational cost. We employ B-spline basis functions as considered in the isogeometric analysis (IGA). We focus on non-stationary Maxwell's equations defined on a regular patch of elements. We employ the idea of alternating-directions splitting (ADS) and employ a second-order accurate time-integration scheme for the time-dependent Maxwell's equations in a weak form. After discretization, the resulting stiffness matrix exhibits a Kronecker product structure. Thus, it enables linear computational cost LU factorization. Additionally, we derive a formulation for absorbing boundary conditions (ABCs) suitable for direction splitting. We perform numerical simulations of the scattering problem (traveling pulse wave) to verify the ABC. We simulate the radiation of electromagnetic (EM) waves from the dipole antenna. We verify the order of the time integration scheme using a manufactured solution problem. We then simulate magnetotelluric measurements. Our simulator is implemented in a shared memory parallel machine, with the GALOIS library supporting the parallelization. We illustrate the parallel efficiency with strong and weak scalability tests corresponding to non-stationary Maxwell simulations.

A NUMERICAL METHOD FOR 3D TIME-DEPENDENT MAXWELL’S EQUATIONS IN AXISYMMETRIC SINGULAR DOMAINS WITH ARBITRARY DATA

In this article, we propose to solve the three-dimensional time-dependent Maxwell equations in a singular axisymmetric domain with arbitrary data. Due to the axisymmetric assumption, the singular computational domain boils down to a subset of R2. However, the electromagnetic field and other vector quantities still belong to R3. Taking advantage that the domain is transformed into a two-dimensional one, by doing Fourier analysis in the third dimension, one arrives to a sequence of singular problems set in a 2D domain. The mathematical tools of such problems have been exposed in [4,19]. Here, we derive a variational method from which we propose an original finite element numerical approach to solve the problem. Numerical experiments are also shown to illustrate that the method is able to capture the singular part of the solution. This approach can also be viewed as a generalization of the Singular Complement Method to three-dimensional problem.

Wellposedness and regularity for linear Maxwell equations with surface current

We study linear time-dependent Maxwell equations on a cuboid consisting of two homogeneous subcuboids. At the interface, we allow for nonzero surface charge density and surface current. This model is a first step towards a detailed mathematical analysis of the interaction of single-layer materials with electromagnetic fields. The main results of this paper provide several wellposedness and regularity statements for the solutions of the Maxwell system. To prove the statements, we employ extension arguments using interpolation theory, as well as semigroup theory and regularity theory for elliptic transmission problems.

Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than H2-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.

A space–time Trefftz DG scheme for the time-dependent Maxwell equations in anisotropic media

In this paper we are concerned with Trefftz discretizations of the time-dependent Maxwell equations in anisotropic media in three-dimensional domains. We propose a class of space–time Trefftz DG methods, which include the Trefftz variational formulation from Egger et al. (2015). We prove the error estimates of the approximate solutions with respect to the meshwidth and the condition number of the coefficient matrices. Furthermore, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous Maxwell equations in anisotropic media. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

Parallel block preconditioners for virtual element discretizations of the time-dependent Maxwell equations

The focus of this study is the construction and numerical validation of parallel block preconditioners for low order virtual element discretizations of the three-dimensional Maxwell equations. The virtual element method (VEM) is a recent technology for the numerical approximation of partial differential equations (PDEs), that generalizes finite elements to polytopal computational grids. So far, VEM has been extended to several problems described by PDEs, and recently also to the time-dependent Maxwell equations. When the time discretization of PDEs is performed implicitly, at each time-step a large-scale and ill-conditioned linear system must be solved, that, in case of Maxwell equations, is particularly challenging, because of the presence of both H(div) and H(curl) discretization spaces. We propose here a parallel preconditioner, that exploits the Schur complement block factorization of the linear system matrix and consists of a Jacobi preconditioner for the H(div) block and an auxiliary space preconditioner for the H(curl) block. Several parallel numerical tests have been performed to study the robustness of the solver with respect to mesh refinement, shape of polyhedral elements, time step size and the VEM stabilization parameter.

Time reversal of surface plasmons

<p style='text-indent:20px;'>We study in this work the so-called "instantaneous time mirrors" in the context of surface plasmons. The latter are associated with high frequency waves at the surface of a conducting sheet. Instantaneous time mirrors were introduced in [<xref ref-type="bibr" rid="b3">3</xref>], with the idea that singular perturbations in the time variable in a wave-type equation create a time-reversed focusing wave. We consider the time-dependent three-dimensional Maxwell's equations, coupled to Drude's model for the description of the surface current. The time mirror is modeled by a sudden, strong, change in the Drude weight of the electrons on the sheet. Our goal is to characterize the time-reversed wave, in particular to quantify the quality of refocusing. We establish that the latter depends on the distance of the source to the sheet, and on some physical parameters such as the relaxation time of the electrons. We also show that, in addition to the plasmonic wave, the time mirror generates a free propagating wave that offers, contrary to the surface wave, some resolution in the direction orthogonal to the sheet. Blurring effects due to non-instantaneous mirrors are finally investigated.</p>

A full-potential and multiscale computational scheme for interactions between ultrafast intense laser pulses and condensed medium

We have presented a multiscale computational scheme for the simulation of interactions between ultrafast intense laser pulses and condensed medium. This approach solves the coupled time-dependent Kohn-Sham (TDKS) and Maxwell equations in real time and provides a unified description of microscopic electron dynamics and macroscopic light propagation from first principles and applicable to wide ultrafast phenomena. Especially, as the TDKS equation was solved within the framework of all-electron full-potential and the augmented plane wave with local orbitals basis sets were adopted, this approach overcomes the drawbacks of real space grids and pseudopotential approximations and works well for some extreme conditions that the information near the nucleus is important. We demonstrated the approach by calculating the microscopic electron dynamics of crystalline silicon (Si) excited with an intense laser pulse and the macroscopic propagation of a laser pulse in a ∼3.5 μm free standing Si film with the feedback of microscopic electron dynamics coupled together. We also verified the availability of this approach under extreme condition with hundreds TPa (1 TPa = 1012 Pa) pressure in a metallic solid neon, and reasonable results were obtained. Our approach provides a new paradigm for multiscale simulations of ultrafast phenomena that induced by intense laser pulses in condensed medium.

Simulation of Maxwell equation based on an ADI approach and integrated radial basis function-generalized moving least squares (IRBF-GMLS) method with reduced order algorithm based on proper orthogonal decomposition

Basic equations of electromagnetic are Maxwell equations. In this manuscript, ADI-IRBF-GMLS is employed for solving the time-dependent Maxwell equations in two-dimension. For approximating the time variable, we utilize alternative direction implicit (ADI) method and integrated radial basis function based on generalized moving least squares (IRBF-GMLS) method is used for space direction. Alternative Direct Implicit (ADI) technique includes two steps in each time stage, that their computations are simple. We have to increase the number of collocation points and also time steps to reach the final time. This procedure increases the used execution time. To overcome this issue, we employ the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure can be called ADI-IRBF-GMLS-POD method. Numerical results are presented and they illustrate the accuracy and efficiency of the proposed method. The point to note is that the used time-discrete scheme i.e. ADI approach cannot be employed in the numerical simulations on non-rectangular computational domains for solving Maxwell equations. This is the main issue in this numerical approach for Maxwell equations. We also compare the ADI-IRBF-GMLS with POD with full model of presented method applied to solve Maxwell equations.

Discontinuous Galerkin Methods with Time-Operators in Their Numerical Traces for Time-Dependent Electromagnetics

Abstract We present a new class of discontinuous Galerkin methods for the space discretization of the time-dependent Maxwell equations whose main feature is the use of time derivatives and/or time integrals in the stabilization part of their numerical traces. These numerical traces are chosen in such a way that the resulting semidiscrete schemes exactly conserve a discrete version of the energy. We introduce four model ways of achieving this and show that, when using the mid-point rule to march in time, the fully discrete schemes also conserve the discrete energy. Moreover, we propose a new three-step technique to devise fully discrete schemes of arbitrary order of accuracy which conserve the energy in time. The first step consists in transforming the semidiscrete scheme into a Hamiltonian dynamical system. The second step consists in applying a symplectic time-marching method to this dynamical system in order to guarantee that the resulting fully discrete method conserves the discrete energy in time. The third and last step consists in reversing the above-mentioned transformation to rewrite the fully discrete scheme in terms of the original variables.

Spheroidal force-free neutron star magnetospheres

Context. Fast rotating and self-gravitating astrophysical objects suffer strong deformations from centrifugal forces. If, moreover, they are magnetised, they generate an electromagnetic wave that is perturbed accordingly. When stellar objects are also surrounded by an ideal plasma, a magnetosphere is formed. For neutron stars, a relativistic magnetised wind is launched, efficiently extracting rotational kinetic energy flowing into particle creation and radiation. Aims. We study the electromagnetic configuration of a force-free magnetosphere encompassing an ideal spheroidal rotating conductor as an inner boundary. We put special emphasis on millisecond period neutron star magnetospheres, that is those showing a significant oblate shape. For completeness, we also investigate the effect of prolate stars. Methods. Force-free solutions were computed by numerical integration of the time-dependent Maxwell equations in spheroidal coordinates using pseudo-spectral techniques. Relevant quantities such as the magnetic field structure, the spin-down luminosity, the polar cap rims, and the current density are shown and compared to the force-free spherical star results. Results. We find that the force-free magnetic field produced by spheroidal stars remains very similar to their spherical counterpart. However, the spin-down luminosity slightly decreases with increasing oblateness or prolateness. Moreover, the polar cap area increases and, for the most part, always encompasses the equivalent spherical star polar cap rims. The polar cap current density is also drastically affected. Conclusions. Neutron stars are significantly distorted by either rotational effects such as millisecond pulsars or by magnetic pressure such as magnetars and high magnetic field pulsars. An observational interpretation of and fitting a thermal X-ray pulsation will greatly benefit from an accurate and quantitative analysis similar to the one presented in this paper. However, even for the fastest possible rotation, the effect would certainly be unobservable, in the sense that we cannot predict what feature of the light curve would supposedly reveal the neutron star deformation due to fast rotation.

A New Block-Diagonal Preconditioner for a Class of $$3\times 3$$ Block Saddle Point Problems

We study the performance of a new block preconditioner for a class of \(3\times 3\) block saddle point problems which arise from finite-element methods for solving time-dependent Maxwell equations and some other practical problems. We also estimate the lower and upper bounds of eigenvalues of the preconditioned matrix. Finally, we examine our new preconditioner to accelerate the convergence speed of the GMRES method which shows the effectiveness of the preconditioner.

Spheroidal magnetic stars rotating in vacuum

Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force, magnetic pressure, or anisotropic equations of state. The stellar surface therefore slightly or significantly deviates from a sphere depending on the strength of these anisotropic perturbations. Aims. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. Methods. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. The solutions are expanded in series of radial and angular spheroidal wave functions. Particular emphasis is put on the magnetic dipole radiation. Second, we compute approximate solutions by integrating the time-dependent Maxwell equations in spheroidal coordinates numerically. Results. We show that the spin-down luminosity corrections compared to a perfect sphere are, to leading order, given by terms involving (a/rL)2 and (a/R)2 where a is the stellar oblateness or prolateness, R the smallest star radius, and rL the light-cylinder radius. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. At large distances r ≫ a, the solution tends asymptotically to the perfect spherical case of a rotating dipole.

Virtual elements for Maxwell's equations

We present a low order virtual element discretization for time dependent Maxwell's equations, which allow for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them on a set of numerical experiments. As pivot results, we discuss some novel inequalities associated with de Rahm sequences of nodal, edge, and face virtual element spaces.

A time-domain finite element scheme and its analysis for nonlinear Maxwell's equations in Kerr media

The purpose of this paper is to develop and analyze a finite element method for solving the time-dependent Maxwell's equations in nonlinear Kerr media. The proposed fully-discrete scheme is proved to be conditionally stable and optimally convergent in the spatial variable and second order in time. Numerical results are presented to support our theoretical analysis and also to demonstrate the practical soliton propagation phenomena in Kerr media.

An arbitrary-order Cell Method with block-diagonal mass-matrices for the time-dependent 2D Maxwell equations

We introduce a new numerical method for the time-dependent Maxwell equations on unstructured meshes in two space dimensions. This relies on the introduction of a new mesh, which is the barycentric-dual cellular complex of the starting simplicial mesh, and on approximating two unknown fields with integral quantities on geometric entities of the two dual complexes. A careful choice of basis functions yields cheaply invertible block-diagonal system matrices for the discrete time-stepping scheme. The main novelty of the present contribution lies in incorporating arbitrary polynomial degree in the approximating functional spaces, defined through a new reference cell. The presented method, albeit a kind of Discontinuous Galerkin approach, requires neither the introduction of user-tuned penalty parameters for the tangential jump of the fields, nor numerical dissipation to achieve stability. In fact an exact electromagnetic energy conservation law for the semi-discrete scheme is proved and it is shown on several numerical tests that the resulting algorithm provides spurious-free solutions with the expected order of convergence.

Analysis and Application of Single Level, Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell's Equations with Random Inputs

Analysis and Application of Single Level, Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell's Equations with Random Inputs

A note on preconditioning for the 3 × 3 block saddle point problem

In this paper, we mainly propose some new preconditioners for a special class of 3 × 3 block saddle point problems, which can arise from the time-dependent Maxwell equations and some other practical problems. Firstly, the solvability of this kind of problem is investigated under suitable assumptions. Then, we analyze the corresponding eigenvalues of the new preconditioned matrices presented in this work. Furthermore, we show that proposed preconditioned matrices only have at most three distinct eigenvalues. Finally, two numerical examples are also carried to verify the effectiveness of proposed preconditioners.