Numerical modeling of an optical beam-splitter by the time-domain wavelet-collocation technique

In this paper, we propose a new time-domain finite element method for solving the time dependent Maxwell's equations coupled with the Lorentz metamaterial model. The Lorentz metamaterial Maxwell's equations are much more complicated than the standard Maxwell's equations in free space. Our fully discrete scheme uses edge elements to approximate the unknowns in space, and uses the leap-frog scheme in time discretization. Numerical stability and the optimal error estimate in the $$L^2$$L2 norm are proved for our proposed scheme. Extensive numerical results are presented to confirm the theoretical analysis and applications of our scheme to model many interesting phenomena happened when wave propagates in the Lorentz metamaterials. Examples include the convergence effect happened in the concave lenses formed by the negative refraction index metamatrials, and total reflection and total transmission observed in the zero index metamaterials.

Hybrid finite element (FE) solutions of transient electromagnetic field are proposed. Time dependent Maxwell's curl equations are solved separately using edge elements for the electric field approximation and nodal-based elements for the magnetic field approximation. Two different formulations are presented: the first adopts a unique mesh, the second two complementary FE meshes. The solution of the resulting equations systems is performed by explicit methods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

We present an efficient method for computing numerically the solution to the time- dependent Maxwell equations in an axisymmetric domain, with arbitrary (not necessarily ax- isymmetric) data. The method is an extension of those introduced in (20) for Poisson's equation, and in (4) for Maxwell's equations in the fully axisymmetric setting (i.e., when the data is also axisymmetric). It is based on a Fourier expansion in the azimuthal direction, and on an improved variant of the Singular Complement Method in the meridian section. When solving Maxwell's equations, this method relies on continuous approximations of the fields, and it is both H(curl)- and H(div)-conforming. Also, it can take into account the lack of regularity of the solution when the domain features non-convex edges or vertices. Moreover, it can handle noisy or ap- proximate data which fail to satisfy the continuity equation, by using either an elliptic correction method or a mixed formulation. We give complete convergence analyses for both mixed and non-mixed formulations. Neither refinements near the reentrant edges or vertices of the domain, nor cutoff functions are required to achieve the desired convergence order in terms of the mesh size, the time step and the number of Fourier modes used.

Finite-element method for time-dependent Maxwell's equations based on an explicit-magnetic-field scheme

A high‐order, leapfrog integrator is presented and analyzed for Maxwell's time domain equations. The integrator is shown to be quite robust in terms of its efficiency and memory demands. Moreover, the integrator retains the original simplicity of the traditional second‐order, leapfrog integrator. Rigorous Fourier analyses are presented to quantify the dispersion, dissipation, and stability properties of the integrator. To limit the scope of the presentation, the 2×2, 2×4, 4×2, and 4×4 schemes are examined and compared. Here the first digit denotes temporal accuracy; the second digit denotes spatial accuracy. Numerical demonstrations, using the three‐dimensional rectangular waveguide as the object under test, are provided to further substantiate the theoretical analysis.

In this paper, we consider the time-dependent Maxwell's equations when Cole–Cole dispersive medium is involved. The Cole–Cole model contains a fractional time derivative term, which couples with the standard Maxwell's equations in free space and creates some challenges in developing and analyzing time-domain finite element methods for solving this model as mentioned in our earlier work [J. Li, J. Sci. Comput., 47 (2001), pp. 1–26]. By adopting some techniques developed for the fractional diffusion equations [V.J. Ervin, N. Heuer, and J.P. Roop, SIAM J. Numer. Anal., 45 (2007), pp. 572–591], [Y. Lin and C. Xu, J. Comput. Phys., 225 (2007), pp. 1533–1552], [F. Liu, P. Zhuang, V. Anh, I. Turner, and K. Burrage, Appl. Math. Comput., 191 (2007), pp. 12–20], we propose two fully discrete mixed finite element schemes for the Cole–Cole model. Numerical stability and optimal error estimates are proved for both schemes. The proposed algorithms are implemented and detailed numerical results are provided to justify our theoretical analysis.

A Fourier penalty method for solving the time-dependent Maxwell's equations in domains with curved boundaries

A novel method of calculating static derivatives and acceleration derivatives using Computational Fluid Dynamics is presented. This method uses field velocity approach to simulate the unsteady forced-plunging flow field over the aircraft without updating the field grids during each time step. This provides an improvement in computational efficiency over similar precision without loss of generality. Finally, to verify the present method, the Basic Finner missile as the benchmark example was taken to calculate its acceleration derivatives at various Mach number. Results are in fairly good agreement with traditionally method and the tunnel experimental results. Comparisons with the results of time-dependent simulations are also included.

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

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 Reflectionless Sponge Layer Absorbing Boundary Condition for the Solution of Maxwell's Equations with High-Order Staggered Finite Difference Schemes

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 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.

The time‐dependent Maxwell equations are one of the most important approaches to describing dynamic or wide‐band frequency electromagnetic phenomena. A sequential finite‐volume, characteristic‐based procedure for solving the time‐dependent, three‐dimensional Maxwell equations has been successfully implemented in Fortran before. Due to its need for a large memory space and high demand on CPU time, it is impossible to test the code for a large array. Hence, it is essential to implement the code on a parallel computing system. In this paper, we discuss an efficient and scalable parallelization of the sequential Fortran time‐dependent Maxwell equations solver using High Performance Fortran (HPF). The background to the project, the theory behind the efficiency being achieved, the parallelization methodologies employed and the experimental results obtained on the Cray T3E massively parallel computing system will be described in detail. Experimental runs show that the execution time is reduced drastically through parallel computing. The code is scalable up to 98 processors on the Cray T3E and has a performance similar to that of an MPI implementation. Based on the experimentation carried out in this research, we believe that a high‐level parallel programming language such as HPF is a fast, viable and economical approach to parallelizing many existing sequential codes which exhibit a lot of parallelism. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Continuous Galerkin methods for solving the time-dependent Maxwell equations in 3D geometries