Flux-splitting dispersion-relation-preserving dual-compact upwind scheme for solving the Maxwell’s equations on non-staggered grids

This study proposes a flux-splitting Maxwell’s equations solver for modeling electromagnetic waves in two-dimensional non-staggered grids. A fifth-order spatially accurate dual-compact upwind scheme was developed in a three-point grid stencil to approximate the first-order derivative term. The integrity of the proposed finite-difference time-domain method for solving TM-mode Maxwell’s equations verified using two-dimensional test problems. The benchmark Mie-scattering problem was also shown to be in good agreement with the semi-analytic result.     

Analysis of microwave induced natural convection in a single mode cavity (Influence of sample volume, placement, and microwave power level)

The heating of water layer using microwave oven with a rectangular waveguide has been studied both numerically and experimentally. The mathematical model is validated with the experimental data. The transient Maxwell’s equations are solved by using the Finite Difference Time Domain (FDTD) method to describe the electromagnetic field inside the waveguide and sample. The temperature profile and velocity field within sample are determined by the solutions of the momentum, energy and Maxwell’s equations. In this study, the effects of physical parameters, e.g. microwave power level, placement of sample inside the waveguide, volume of sample, are studied. The distribution of electric field, temperature profile and velocity field are presented in details. The results show good agreement between simulation results and experimental data. Conclusively, the mathematical model presented here correctly explains the phenomena of microwave heating of water layer.     

Numerical analysis of a PML model for time-dependent Maxwell’s equations

In this paper, we study a perfectly matched layer model for the three-dimensional time-dependent Maxwell’s equations. We develop both semi- and fully-discrete finite element methods for solving the truncated PML problem by Nedelec edge elements. Optimal convergence rates are proved for both semi- and fully-discrete schemes. To our knowledge, this is the first error analysis obtained for time domain finite element method for PML models.     

UPML formulation for truncating conductive media in curvilinear coordinates

This work presents a formulation based on UPML for truncating conductive media by using a local and non-orthogonal coordinate system to solve Maxwell’s equations by the FDTD method. The detailed procedure for obtaining the UPML equations for this case is shown and the complete equation set is provided.     

Numerical electroseismic modeling: A finite element approach

Electroseismics is a procedure that uses the conversion of electromagnetic to seismic waves in a fluid-saturated porous rock due to the electrokinetic phenomenon. This work presents a collection of continuous and discrete time finite element procedures for electroseismic modeling in poroelastic fluid-saturated media. The model involves the simultaneous solution of Biot’s equations of motion and Maxwell’s equations in a bounded domain, coupled via an electrokinetic coefficient, with appropriate initial conditions and employing absorbing boundary conditions at the artificial boundaries. The 3D case is formulated and analyzed in detail including results on the existence and uniqueness of the solution of the initial boundary value problem. Apriori error estimates for a continuous-time finite element procedure based on parallelepiped elements are derived, with Maxwell’s equations discretized in space using the lowest order mixed finite element spaces of Nédélec, while for Biot’s equations a nonconforming element for each component of the solid displacement vector and the vector part of the Raviart-Thomas-Nédélec of zero order for the fluid displacement vector are employed. A fully implicit discrete-time finite element method is also defined and its stability is demonstrated. The results are also extended to the case of tetrahedral elements. The 2D cases of compressional and vertically polarized shear waves coupled with the transverse magnetic polarization (PSVTM-mode) and horizontally polarized shear waves coupled with the transverse electric polarization (SHTE-mode) are also formulated and the corresponding finite element spaces are defined. The 1D SHTE initial boundary value problem is also formulated and approximately solved using a discrete-time finite element procedure, which was implemented to obtain the numerical examples presented.     

Fully discrete linear approximation scheme for electric field diffusion in type-II superconductors

Superconductors are attracting physicists thanks to their ability to conduct electric current with virtually zero resistance. Their nonlinear behaviour opens, on the other hand, challenging problems for mathematicians. Our model of the diffusion of electric field in superconductors is based on three pillars: the eddy-current version of Maxwell’s equations, power law model of type-II superconductivity and linear dependence of magnetic induction on magnetic field. This leads to a time-dependent nonlinear degenerate partial differential equation. We propose a linear fully discrete approximation scheme to solve it. We have proven the convergence of the method and derived the error estimates describing the dependence of the error on the discretization parameters. These theoretical results were successfully confronted with numerical experiments.     

Low order Crouzeix-Raviart type nonconforming finite element methods for the 3D time-dependent Maxwell’s equations

Two Crouzeix-Raviart type nonconforming elements are used in a finite element scheme as well in a mixed finite element scheme for time-dependent Maxwell’s equations in three dimensions. The error estimates are obtained under anisotropic meshes, which are the same as those for conforming elements under regular meshes.     

Solitary waves for the Klein-Gordon-Maxwell system with critical exponent

In this paper we study the existence of solutions for nonlinear Klein-Gordon-Maxwell equations coupled with Maxwell’s equations when the nonlinearity exhibits critical growth. We improve some previous existence results in Azzollini et al. (2009) [5], Carrião et al. (2009) [4] and Cassani (2004) [3].     

Reconstruction of closely spaced small inhomogeneities via boundary measurements for the full time-dependent Maxwell’s equations

We consider for the full time-dependent Maxwell’s equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic boundary measurements on the boundary for a finite time interval.     

Three-dimensional invisible cloaks with arbitrary shapes based on partial differential equation

The method of designing electromagnetic invisible cloak is usually based on the form-invariance of Maxwell’s equations in coordinate transformation. By solving the partial differential equations (PDEs) that describe how the coordinates transform, three-dimensional (3-D) electromagnetic and acoustic invisible cloaks with arbitrary shapes can be designed provided the boundary conditions of the cloaks can be determined by the corresponding transformation. Full wave simulations based on finite element method verify the designed cloaks. The proposed method can be easily used in designing other transformation media such as matter-wave cloaks.     

Dielectric waveguides of arbitrary cross sectional shape

Numerical guided mode solutions of arbitrary cross sectional shaped waveguides are obtained using a finite difference (FD) technique. The standard FD scheme is appropriately modified to capture all discontinuities, due to the change of the refractive index, across the waveguides’ interfaces taking into account the shape of each interface at the same time. The method is applied to the vector Helmholtz equation formulated to describe the electric or magnetic fields in the waveguide (one field is retrieved from the other through Maxwell’s equations). Computational cost is kept to a minimum by exploiting sparse matrix algebra. The waveguides under study have arbitrary cross sectional shape and arbitrary refractive index profile.     

A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics

In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell’s equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leap-frog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method.     

A simulation code for investigating 2D heating of material bodies by low frequency electric fields

The finite difference method has been used to simultaneously solve in two dimensions Maxwell's equations and the heat transfer equation in forms which are appropriate to modelling low frequency electrical heating of solid materials. The nonlinear coupling of these modelling equations, which is due to temperature dependent electrical conductivities, necessitates the use of an explicit-sequential solution method and the limiting of the timestep size to ensure stability. The finite difference equations were modified to account for sharp electrical conductivity differences between different media in the body being heated.The simulation code was tested by comparison of the simulator predictions with the measured results of a physical scale model experiment. The simulation code was able to accurately predict the resistance between the electrodes used for heating, the energy deposition and the temperature rise in the bulk of the physical model.     

A non-overlapping domain decomposition for low-frequency time-harmonic Maxwell’s equations in unbounded domains

In this paper, we are concerned with a non-overlapping domain decomposition method for solving the low-frequency time-harmonic Maxwell’s equations in unbounded domains. This method can be viewed as a coupling of finite elements and boundary elements in unbounded domains, which are decomposed into two subdomains with a spherical artificial boundary. We first introduce a discretization for the coupled variational problem by combining Nédélec edge elements of the lowest order and curvilinear elements. Then we design a D-N alternating method for solving the discrete problem. In the method, one needs only to solve the finite element problem (in a bounded domain) and calculate some boundary integrations, instead of solving a boundary integral equation. It will be shown that such iterative algorithm converges with a rate independent of the mesh size. The work of Qiya Hu was supported by Natural Science Foundation of China G10371129.     

Numerical solutions of nonlinear evolution equations using variational iteration method

The variational iteration method is used to solve three kinds of nonlinear partial differential equations, coupled nonlinear reaction diffusion equations, Hirota–Satsuma coupled KdV system and Drinefel’d–Sokolov–Wilson equations. Numerical solutions obtained by the variational iteration method are compared with the exact solutions, revealing that the obtained solutions are of high accuracy. He's variational iteration method is introduced to overcome the difficulty arising in calculating Adomian polynomial in Adomian method. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Twisted dendriform algebras and the pre-Lie Magnus expansion

In this paper an application of the recently introduced pre-Lie Magnus expansion to Jackson’s q-integral and q-exponentials is presented. Twisted dendriform algebras, which are the natural algebraic framework for Jackson’s q-analogs, are introduced for that purpose. It is shown how the pre-Lie Magnus expansion is used to solve linear q-differential equations. We also briefly outline the theory of linear equations in twisted dendriform algebras.     

On the coupling of LDG-FEM and BEM methods for the three dimensional magnetostatic problem

We present two new coupling models for the three dimensional magnetostatic problem. In the first model, we propose a new coupled formulation, prove that it is well posed and solves Maxwell’s equations in the whole space. In the second, we propose a new coupled formulation for the Local Discontinuous Galerkin method, the finite element method and the boundary element method. This formulation is obtained by coupling the LDG method inside a bounded domain Ω₁ with the FEM method inside a layer where Ω is a bounded domain which is made up of material of permeability μ and such that , and with a boundary element method involving Calderon’s equations. We prove that our formulation is consistent and well posed and we present some a priori error estimates for the method.     

Approximate analytical solutions for Kolmogorov’s equations

This paper reports the explicit analytical solutions for Kolmogorov’s equations. Kolmogorov’s equations are commonly used to describe the structure of local isotropic turbulence, but their exact analytical solutions have not yet been found. In this paper, the closed-form solutions for two kinds of Kolmogorov’s equations are obtained. The derivations of the approximate solutions are based on the homotopy analysis method, which is a new tool for obtaining the approximate analytical solutions of both strong and weak nonlinear differential equations. To examine the validity of the approximate solutions, numerical comparisons between results from the homotopy analysis method and the fourth-order Runge-Kutta method are carried out. It is shown that the results are in good agreement.     

On homotopy analysis method applied to linear optimal control problems

In this paper, the homotopy analysis method (HAM) is employed to solve the linear optimal control problems (OCPs), which have a quadratic performance index. The study examines the application of the homotopy analysis method in obtaining the solution of equations that have previously been obtained using the Pontryagin’s maximum principle (PMP). The HAM approach is also applied in obtaining the solution of the matrix Riccati equation. Numerical results are presented for several test examples involving scalar and 2nd-order systems to demonstrate the applicability and efficiency of the method.     

A new scheme for the solution of reaction diffusion and wave propagation problems

In this paper, a robust numerical scheme is presented for the reaction diffusion and wave propagation problems. The present method is rather simple and straightforward. The Houbolt method is applied so as to convert both types of partial differential equations into an equivalent system of modified Helmholtz equations. The method of fundamental solutions is then combined with the method of particular solution to solve these new systems of equations. Next, based on the exponential decay of the fundamental solution to the modified Helmholtz equation, the dense matrix is converted into an equivalent sparse matrix. Finally, verification studies on the sensitivity of the method’s accuracy on the degree of sparseness and on the time step magnitude of the Houbolt method are carried out for four benchmark problems.