Convergent geometric integrator for the Landau-Lifshitz-Gilbert equation in micromagnetics

We consider a lowest-order finite element scheme for the Landau-Lifshitz-Gilbert equation (LLG) which describes the dynamics of micromagnetism. In contrast to previous works, we examine LLG with a total magnetic field which is induced by several physical phenomena described in terms of exchange energy, anisotropy energy, magnetostatic energy, and Zeeman energy. In our numerical scheme, the highest-order term which stems from the exchange energy, is treated implicitly, whereas the remaining energy contributions are computed explicitly. Therefore, only one sparse linear system has to be solved per time-step. The proposed scheme is unconditionally convergent to a global weak solution of LLG. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An iterative algorithm for a Cauchy problem in eddy-current modelling

We consider a linear steady-state eddy-current problem for a magnetic field in a bounded domain. The boundary consists of two parts: reachable with prescribed Cauchy data and unreachable with no data on it. We design an iterative (Landweber type) algorithm for solution of this problem. At each iteration step two auxiliary mixed well-posed boundary value problems are solved. The analysis of temporary problems is performed in suitable function spaces. This creates the basis for the convergence argument. The theoretical results are supported with numerical experiments.     

Solving semidefinite programming problems via alternating direction methods

In this paper, we consider an alternating direction algorithm for the solution of semidefinite programming problems (SDP). The main idea of our algorithm is that we reformulate the complementary conditions in the primal–dual optimality conditions as a projection equation. By using this reformulation, we only need to make one projection and solve a linear system of equation with reduced dimension in each iterate. We prove that the generated sequence converges to the solution of the SDP under weak conditions.     

Weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics

We consider the time-dependent Landau–Lifshitz–Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii–Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weak–strong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.     

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.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Differential variational–hemivariational inequalities with application to contact mechanics

In this paper we study a dynamical system which consists of the Cauchy problem for a nonlinear evolution equation of first order coupled with a nonlinear time-dependent variational–hemivariational inequality with constraint in Banach spaces. The evolution equation is considered in the framework of evolution triple of spaces, and the inequality which involves both the convex and nonconvex potentials. We prove existence of solution by the Kakutani–Ky Fan fixed point theorem combined with the Minty formulation and the theory of hemivariational inequalities. We illustrate our findings by examining a nonlinear quasistatic elastic frictional contact problem for which we provide a result on existence of weak solution.     

Adaptive operator splitting finite element method for Allen–Cahn equation

In this paper, a new numerical method is proposed and analyzed for the Allen–Cahn (AC) equation. We divide the AC equation into linear section and nonlinear section based on the idea of operator splitting. For the linear part, it is discretized by using the Crank–Nicolson scheme and solved by finite element method. The nonlinear part is solved accurately. In addition, a posteriori error estimator of AC equation is constructed in adaptive computation based on superconvergent cluster recovery. According to the proposed a posteriori error estimator, we design an adaptive algorithm for the AC equation. Numerical examples are also presented to illustrate the effectiveness of our adaptive procedure.     

Fractional-step <Emphasis Type="Italic">θ</Emphasis>-method for solving singularly perturbed problem in ecology

We consider a predator-prey model arising in ecology that describes a slow-fast dynamical system. The dynamics of the model is expressed by a system of nonlinear differential equations having different time scales. Designing numerical methods for solving problems exhibiting multiple time scales within a system, such as those considered in this paper, has always been a challenging task. To solve such complicated systems, we therefore use an efficient time-stepping algorithm based on fractional-step methods. To develop our algorithm, we first decouple the original system into fast and slow sub-systems, and then apply suitable sub-algorithms based on a class of θ-methods, to discretize each sub-system independently using different time-steps. Then the algorithm for the full problem is obtained by utilizing a higher-order product method by merging the sub-algorithms at each time-step. The nonlinear system resulting from the use of implicit schemes is solved by two different nonlinear solvers, namely, the Jacobian-free Newton-Krylov method and the well-known Anderson’s acceleration technique. The fractional-step θ-methods give us flexibility to use a variety of methods for each sub-system and they are able to preserve qualitative properties of the solution. We analyze these methods for stability and convergence. Several numerical results indicating the efficiency of the proposed method are presented. We also provide numerical results that confirm our theoretical investigations.     

A numerical scheme based on a solution of nonlinear advection–diffusion equations

A mathematical formulation of the two-dimensional Cole–Hopf transformation is investigated in detail. By making use of the Cole–Hopf transformation, a nonlinear two-dimensional unsteady advection–diffusion equation is transformed into a linear equation, and the transformed equation is solved by the spectral method previously proposed by one of the authors. Thus a solution to initial value problems of nonlinear two-dimensional unsteady advection–diffusion equations is derived. On the base of the solution, a numerical scheme explicit with respect to time is presented for nonlinear advection–diffusion equations. Numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality.     

New Tchebyshev‐Galerkin operational matrix method for solving linear and nonlinear hyperbolic telegraph type equations

The telegraph equation describes various phenomena in many applied sciences. We propose two new efficient spectral algorithms for handling this equation. The principal idea behind these algorithms is to convert the linear/nonlinear telegraph problems (with their initial and boundary conditions) into a system of linear/nonlinear equations in the expansion coefficients, which can be efficiently solved. The main advantage of our algorithm in the linear case is that the resulting linear systems have special structures that reduce the computational effort required for solving them. The numerical algorithms are supported by a careful convergence analysis for the suggested Chebyshev expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithms. © 2016Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1553–1571, 2016     

A hard ferromagnetic and elastic beam-plate sandwich structure

In this paper, the deformation of a composite hard ferromagnetic-elastic beam-plate structure is investigated. A sandwich structure, composed of two thin hard ferromagnetic layers, with a linear elastic layer in between, is considered. The deformation is due to the self generated magnetic field (magnetostriction). The aim is to assess the interaction forces among the perfectly bonded layers, through a consistent application of the classical nonlinear magneto-elastic theory. Once the general mechanical model is stated, the analysis is specialized to study longitudinal elongation, given its great relevance in technical applications. Owing to the non-local character of the magnetic action, a nonlinear integro-differential equation is derived. Some qualitative properties of the solution are pointed out and the asymptotic behavior near the end sections is examined in detail. A finite differences approach allows writing an approximating nonlinear system of equations in the non asymptotic part of the solution, which is solved through a Newton’s iterative scheme. The numerical results are discussed and it is shown how the asymptotic part of the solution well approximates the full behavior of the structure. Furthermore, the longitudinal interaction force density is found to be singular at the end cross-sections, regardless of the assumed bonding type.     

A hard ferromagnetic and elastic beam-plate sandwich structure

In this paper, the deformation of a composite hard ferromagnetic-elastic beam-plate structure is investigated. A sandwich structure, composed of two thin hard ferromagnetic layers, with a linear elastic layer in between, is considered. The deformation is due to the self generated magnetic field (magnetostriction). The aim is to assess the interaction forces among the perfectly bonded layers, through a consistent application of the classical nonlinear magneto-elastic theory. Once the general mechanical model is stated, the analysis is specialized to study longitudinal elongation, given its great relevance in technical applications. Owing to the non-local character of the magnetic action, a nonlinear integro-differential equation is derived. Some qualitative properties of the solution are pointed out and the asymptotic behavior near the end sections is examined in detail. A finite differences approach allows writing an approximating nonlinear system of equations in the non asymptotic part of the solution, which is solved through a Newton’s iterative scheme. The numerical results are discussed and it is shown how the asymptotic part of the solution well approximates the full behavior of the structure. Furthermore, the longitudinal interaction force density is found to be singular at the end cross-sections, regardless of the assumed bonding type.     

Newton方法在非线性振动理论中的推广与应用

本文提出和证明了,用Newton方法可以求解强(弱)非线性非自治系统的渐近解析周期解,为研究强(弱)非线性系统振动提供了一个新的解析方法.根据本文方法的需要,讨论了二阶线性非齐次周期系统周期解的存在与计算问题.此外,还讨论了Newton方法对于拟线性系统的应用.最后,应用本文方法计算了Duffing方程的周期解.     

Solution of the linear and nonlinear advection–diffusion problems on a sphere

Various linear advection–diffusion problems and nonlinear diffusion problems on a sphere are considered and solved using the direct, implicit and unconditionally stable finite-volume method of second-order approximation in space and time. In the absence of external forcing and dissipation, the method preserves the total mass of the substance and the norm of the solution. The component wise operator splitting allows us to develop the direct (noniterative) and fast numerical algorithm. The split problems in the longitudinal direction are solved using the Sherman-Morrison formula and Thomas algorithm. The direct solution of the split problems in the latitudinal direction requires the use of the bordering method for a block matrix, and the preliminary determination of the solution at the poles. The resulting systems with tridiagonal matrices are solved by the Thomas algorithm. The numerical experiments demonstrate that the method correctly describes the local advection–diffusion processes on the sphere, in particular, through the poles, and accurately simulate blow-up regimes (unlimited growing solutions) of nonlinear combustion, the propagation of nonlinear temperature and spiral waves, and solutions to Gray-Scott reaction–diffusion model.     

Nonlinear resonances of a semi-infinite cable on a nonlinear elastic foundation

The Multiple Time Scale (MTS) method is applied to the study of nonlinear resonances of a semi-infinite cable resting on a nonlinear elastic foundation, subject to a constant uniformly distributed load and to a linear viscous damping force. The zero order solution provides the static displacement, which is governed by a nonlinear equation which has been solved in closed form. The first order solution provides the linear resonances, which are seen to be functions of the nonlinearity parameter and of the static displacement at the finite boundary only. Although the first-order governing equation is linear, it has non constant coefficients and cannot be solved in closed form, so that a numerical solution is considered; the eigenfrequencies obtained in this way are also compared with the approximate eigenvalues obtained by the WKB method. At the second order of the MTS expansion, we see that the solution is independent of the intermediate time scale; some additional terms are present, including a time-independent shift of the average position of the oscillations. Finally, the nonlinear frequency–amplitude response curves, which are investigated in detail and which represent the main result of this work, are obtained from the solvability condition at the third order.     

A Constrained Optimization Approach for LCP

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.     

Expanding stability regions of explicit advective-diffusive finite difference methods by Jacobi preconditioning

Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions that are placed on stable time-step intervals. Stability bounds for explicit time differencing methods on advective–diffusive problems are generally determined by the diffusive part of the problem. These bounds are very small and implicit methods are used instead. The linear systems arising from these implicit methods are generally solved by iterative methods. In this article we develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods. A Jacobi preconditioned time differencing method is defined and analyzed for both diffusion and advection–diffusion equations. Several computational examples of both linear and nonlinear advective-diffusive problems are solved to demonstrate the accuracy and improved stability limits. © 1995 John Wiley & Sons, Inc.     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Solving Second-Order Differential Equations with Lie Symmetries

Lie"s theory for solving second-order quasilinear differential equations based on its symmetries is discussed in detail. Great importance is attached to constructive procedures that may be applied for designing solution algorithms. To this end Lie"s original theory is supplemented by various results that have been obtained after his death one hundred years ago. This is true above all of Janet"s theory for systems of linear partial differential equations and of Loewy"s theory for decomposing linear differential equations into components of lowest order. These results allow it to formulate the equivalence problems connected with Lie symmetries more precisely. In particular, to determine the function field in which the transformation functions act is considered as part of the problem. The equation that originally has to be solved determines the base field, i.e. the smallest field containing its coefficients. Any other field occurring later on in the solution procedure is an extension of the base field and is determined explicitly. An equation with symmetries may be solved in closed form algorithmically if it may be transformed into a canonical form corresponding to its symmetry type by a transformation that is Liouvillian over the base field. For each symmetry type a solution algorithm is described, it is illustrated by several examples. Computer algebra software on top of the type system ALLTYPES has been made available in order to make it easier to apply these algorithms to concrete problems.