Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

Global well-posedness and relaxation limits of a model for radiating gas

We study the initial value problem for a hyperbolic-elliptic coupled system with arbitrary large discontinuous initial data. We prove existence and uniqueness for that model by means of L¹-contraction and comparison properties. Moreover, after suitable scalings, we study both the hyperbolic-parabolic and the hyperbolic-hyperbolic relaxation limits for that system.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

Green's function-based integral approaches to linear transient boundary-value problems and their stability characteristics (I)

Two Green's function-based formulations are applied to the governing differential equation which describes unsteady heat or mass transport in an isotropic homogeneous 1-D domain. In this first part of a two series of papers, the linear form of the differential equation is addressed. The first formulation, herein denoted the quasi-steady Green element (QSGE) formulation, uses the Laplace differential operator as auxiliary equation to obtain the singular integral representation of the governing equation, while the second, denoted the transient Green element (TGE), uses the transient heat equation as auxiliary equation. The mathematical simplicity of the Green's function of the first formulation enhances the ease of solution of the integral equations and the resultant discrete equations. From the point of computational convenience, therefore, the first formulation is preferred. The stability characteristics of the two formulations are evaluated by examining how they propagate various Fourier harmonics in speed and amplitude. We found that both formulations correctly reproduce the theoretical speed of the harmonics, but fail to propagate the amplitude of the small harmonics correctly for Courant value of about unity. The QSGE formulation with difference weighting values between 0.67 and 0.75, and the TGE formulation provide optimal performance in numerical stability.     

Adaptive techniques for Landau–Lifshitz–Gilbert equation with magnetostriction

In this paper we propose a time–space adaptive method for micromagnetic problems with magnetostriction. The considered model consists of coupled Maxwell's, Landau–Lifshitz–Gilbert (LLG) and elastodynamic equations. The time discretization of Maxwell's equations and the elastodynamic equation is done by backward Euler method, the space discretization is based on Whitney edge elements and linear finite elements, respectively. The fully discrete LLG equation reduces to an ordinary differential equation, which is solved by an explicit method, that conserves the norm of the magnetization.     

WAVES IN FERROMAGNETIC MEDIA

ABSTRACT

It is shown that small perturbations of equilibrium states in ferromagnetic media give rise to standing and traveling waves that are stable for long times. The evolution of the wave profiles is governed by semilinear heat equations. The mathematical model underlying these results consists of the Landau–Lifshitz equation for the magnetization vector and Maxwell's equations for the electromagnetic field variables. The model belongs to a general class of hyperbolic equations for vector-valued functions, whose asymptotic properties are analyzed rigorously. The results are illustrated with numerical examples.     

Mildly degenerate Kirchhoff equations with weak dissipation: Global existence and time decay

We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t→+∞.We prove that the hyperbolic problem has a unique global solution for suitable values of the parameters. We also prove that the solution decays to zero, as t→+∞, with the same rate of the solution of the limit problem of parabolic type.     

Two variants of magnetic diffusivity stabilized finite element methods for the magnetic induction equation

We consider the time‐dependent magnetic induction model where the sought magnetic field interacts with a prescribed velocity field. This coupling results in an additional force term and time dependence in Maxwell's equation. We propose two different magnetic diffusivity stabilized continuous nodal‐based finite element methods for this problem. The first formulation simply adds artificial magnetic diffusivity to the partial differential equation, whereas the second one uses a local projected magnetic diffusivity as stabilization. We describe those methods and analyze them semi‐discretized in space to get bounds on stabilization parameters where we distinguish equal‐order elements and Taylor‐Hood elements. Different numerical experiments are performed to illustrate our theoretical findings.     

Stability of transonic shock fronts in two-dimensional Euler systems

We study the stability of stationary transonic shock fronts under two-dimensional perturbation in gas dynamics. The motion of the gas is described by the full Euler system. The system is hyperbolic ahead of the shock front, and is a hyperbolic-elliptic composed system behind the shock front. The stability of the shock front and the downstream flow under two-dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic-elliptic composed system. We develop a method to deal with boundary value problems for such systems. The crucial point is to decompose the system to a canonical form, in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients. By several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem. Our result indicates the stability of the transonic shock front and the flow field behind the shock.

     

Green's function-based integral approaches to nonlinear transient boundary-value problems (II)

Two Green's function-based numerical formulations are used to solve the time-dependent nonlinear heat conduction (diffusion) equation. These formulations, which are an extension of the first paper, utilize two fundamental solutions and the Green's second identity to achieve integral replications of the governing partial differential equation. The integral equations thus derived are discretized in space and time and aggregated in a finite element sense to give a system of nonlinear discrete equations that are solved by the Newton–Raphson algorithm. The mathematical simplicity of the Green's function of the first formulation facilitates its numerical implementation. The performance of the formulations is assessed by comparing their results with available numerical and analytical solutions. In all cases satisfactory and physically realistic results are obtained.     

On a boundary variational inequality of the second kind modelling a friction problem

A scalar contact problem with friction governed by the Yukawa equation is reduced to a boundary variational inequality. The presence of the non‐differentiable friction functional causes some difficulties when approximated. We present two methods to overcome this difficulty. The first one is a regularization leading to a non‐linear boundary variational equation, for which we propose an iterative procedure, whereas the second method is based on the boundary mixed variational formulation involving Lagrange multipliers. We propose Uzawa's algorithm to compute the saddle point of the corresponding boundary Lagrangian and investigate the discretization of various formulations by the boundary element Galerkin method. Convergence of the boundary element solution is proved and a convergence order is obtained. Copyright © 2002 John Wiley & Sons, Ltd.     

Solitary waves in Abelian Gauge Theories with strongly nonlinear potentials

We study the existence of radially symmetric solitary waves for a system of a nonlinear Klein–Gordon equation coupled with Maxwell's equation in presence of a positive mass. The nonlinear potential appearing in the system is assumed to be positive and with more than quadratical growth at infinity.     

Characteristic Boundary Layers for Mixed Hyperbolic-Parabolic Systems in One Space Dimension and Applications to the Navier-Stokes and MHD Equations

We provide a detailed analysis of the boundary layers for mixed hyperbolic-parabolic systems in one space dimension and small amplitude regimes. As an application of our results, we describe the solution of the so-called boundary Riemann problem recovered as the zero viscosity limit of the physical viscous approximation. In particular, we tackle the so-called doubly characteristic case, which is considerably more demanding from the technical viewpoint and occurs when the boundary is characteristic for both the mixed hyperbolic-parabolic system and for the hyperbolic system obtained by neglecting the second-order terms. Our analysis applies in particular to the compressible Navier-Stokes and MHD equations in Eulerian coordinates, with both positive and null electrical resistivity. In these cases, the doubly characteristic case occurs when the velocity is close to 0. The analysis extends to nonconservative systems. © 2020 Wiley Periodicals, Inc.     

The electric field in the conductive half space as a model in mining and petroleum prospecting

We present a formulation of Maxwell's equations in a conductive medium, in the time domain. In order to restrict the equations to the conductive half space, the solution in the air is represented by an integral boundary operator on the interface. The problem admits a variational formulation, allowing a finite element solution. A mathematical analysis is described for 2D and 3D models, and numerical results are presented.     

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

On the one- and one-half-dimensional relativistic Vlasov-Fokker-Planck-Maxwell system

In this paper, we study the relativistic Vlasov-Fokker-Planck-Maxwell system in one space variable and two momentum variables. This non-linear system of equations consists of a transport equation for the phase space distribution function combined with Maxwell's equations for the electric and magnetic fields. It is important in modelling distribution of charged particles in the kinetic theory of plasma. We prove the existence of a classical solution when the initial density decays fast enough with respect to the momentum variables. The solution which shares this same decay condition along with its first derivatives in the momentum variables is unique.     

Analysis of stability and dispersion in a finite element method for Debye and Lorentz dispersive media

We study the stability properties of, and the phase error present in, a finite element scheme for Maxwell's equations coupled with a Debye or Lorentz polarization model. In one dimension we consider a second order formulation for the electric field with an ordinary differential equation for the electric polarization added as an auxiliary constraint. The finite element method uses linear finite elements in space for the electric field as well as the electric polarization, and a theta scheme for the time discretization. Numerical experiments suggest the method is unconditionally stable for both Debye and Lorentz models. We compare the stability and phase error properties of the method presented here with those of finite difference methods that have been analyzed in the literature. We also conduct numerical simulations that verify the stability and dispersion properties of the scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An hp-Efficient Residual-Based A Posteriori Error Estimator for Maxwell's Equations

We present a residual-based a posteriori error estimator for Maxwell's equations in the electric field formulation. The error estimator is formulated in terms of the residual of the considered problem and we state upper and lower bounds in terms of the energy error of the computed solution for the estimator. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.