Reduction of the equations of dynamics of systems with constraints to a given structure

The problem of constructing systems of second-order ordinary differential equations, the solutions of which, with the appropriate initial conditions, satisfy given equations of the constraints, is considered. The conditions for representing the differential equations in the form of Lagrange equations of the second kind are determined. It is shown that, when the equations of the non-holonomic constraints are specified by polynomials of order no higher than two with respect to the generalized velocities, the generalized forces of a system with energy dissipation comprise the sum of the gyroscopic, potential and dissipative forces.     

A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations

In this paper, we consider a class of mathematical programs governed by second-order cone constrained parameterized generalized equations. We reformulate the necessary optimality conditions as a system of nonsmooth equations under linear independence constraint qualification and the strict complementarity condition. A set of second order sufficient conditions is proposed, which is proved to be sufficient for the second order growth of the stationary point. The smoothing Newton method in [40] is employed to solve the system of nonsmooth equations whose strongly BD-regularity at a solution point is demonstrated under the second order sufficient conditions. Several illustrative examples are provided and discussed.     

An orthogonal spline collocation alternating direction implicit method for second-order hyperbolic problems

On a rectangular region, we consider a linear second-order hyperbolicinitial-boundary value problem involving a mixed derivativeterm, continuous variable coefficients and non-homogeneous Dirichletboundary conditions. In comparison to the alternating directionimplicit Laplace-modified method of Fernandes (1997), we formulateand analyse a new parameter-free alternating direction implicitscheme in which the standard central difference formula is usedfor the time approximation and orthogonal spline collocationis used for the spatial discretization. We establish unconditionalstability of the scheme, and its optimal order in the discretemaximum norm in time and the H¹ norm in space. Numerical experimentsindicate that the new scheme, which has the same order as themethod of Fernandes (1997, Numer. Math., 77, 223–241),is more accurate. We also show that the new scheme is easilygeneralized to the second-order hyperbolic problems on rectangularpolygons. Extensions of the scheme to problems with discontinuouscoefficients, nonlinear problems, and problems with other boundaryconditions are also discussed.     

Construction and asymptotic stability of structurally stable internal layer solutions

We introduce a geometric/asymptotic method to treat structurally stable internal layer solutions. We consider asymptotic expansions of the internal layer solutions and the critical eigenvalues that determine their stability. Proofs of the existence of exact solutions and eigenvalue-eigenfunctions are outlined.

Multi-layered solutions are constructed by a new shooting method through a sequence of pseudo Poincaré mappings that do not require the transversality of the flow to cross sections. The critical eigenvalues are determined by a coupling matrix that generates the SLEP matrix. The transversality of the shooting method is related to the nonzeroness of the critical eigenvalues.

An equivalent approach is given to mono-layer solutions. They can be determined by the intersection of a fast jump surface and a slow switching curve, which reduces Fenichel's transversality condition to the slow manifold. The critical eigenvalue is determined by the angle of the intersection.

We present three examples. The first treats the critical eigenvalues of the system studied by Angenent, Mallet-Paret & Peletier. The second shows that a key lemma in the SLEP method may not hold. The third is a perturbed activator-inhibitor system that can have any number of mono-layer solutions. Some of the solutions can only be found with the new shooting method.

     

Non-variational approximation of discrete eigenvalues of self-adjoint operators

^** Email: L.Boulton{at}ma.hw.ac.uk We establish sufficient conditions for approximation of discreteeigenvalues of self-adjoint operators in the second-order projectionmethod suggested recently in Levitin & Shargorodsky (2004,Spectral pollution and second order relative spectra for self-adjointoperators. IMA J. Numer. Anal., 24, 393–416). We findfairly explicit estimates for the eigenvalue error and studyin detail two concrete model examples. Our results show thatsecond-order projection strategies not only are universallypollution free but also achieve approximation under naturalconditions on the discretising basis.     

Linear differential operators and operator matrices of the second order

Linear differential operators (equations) of the second order in Banach spaces of vector functions defined on the entire real axis are studied. Conditions of their invertibility are given. The main results are based on putting a differential operator in correspondence with a second-order operator matrix and further use of the theory of first-order differential operators that are defined by the operator matrix. A general scheme is presented for studying the solvability conditions for different classes of second-order equations using second-order operator matrices. The scheme includes the studied problem as a special case.     

Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions

The elastic stiffness or compliance is a fourth-order tensorthat can be expressed in terms of two second-order symmetrictensors A and B and a fourth-order completely symmetric andtraceless tensor Z (or z). It is shown that the parts associatedwith A, B and Z (or z) are all structurally invariant undera three-dimensional transformation. Thus a linear combinationof the three parts gives a general expression for three-dimensionalstructural invariants. All three-dimensional structural invariantsavailable in the literature are shown to be special cases ofthis general expression. Invariants that are inherited by eachstructural invariant are presented.     

直立圆柱二阶波浪力解析解

大直径直立圆柱体上的二阶波浪力目前已有一些研究结果,但仍存在一些值得进一步探讨之处.这一方面在于二阶辐射条件还不甚清楚:另一方面在于已有的二阶力公式或是所含积分的收敛精度不易保证,或是表达式繁杂,不利于实际计算.本文在求解这一问题时,不是对二阶势提出辐射条件,而是对二阶势的周向富里叶分量提出辐射条件──Sommerfeld辐射条件.求解中,利用本文推导出的数学公式,简化了二阶自由面条件非齐次项的表达式,得到了形式简单,易于计算的二阶波浪力精确公式.二阶力计算结果与实验结果吻合良好.     

带非线性源项的Riesz回火分数阶扩散方程的预估校正方法

本文针对带非线性源项的Riesz回火分数阶扩散方程,利用预估校正方法离散时间偏导数,并用修正的二阶Lubich回火差分算子逼近Riesz空间回火的分数阶偏导数,构造出一类新的数值格式.给出了数值格式在一定条件下的稳定性与收敛性分析,且该格式的时间与空间收敛阶均为二阶.数值试验表明数值方法是有效的.     

Second-order conditions in c1, 1 optimization with applications

Second-order necessary conditions for inequality and equality constrained C^{1, 1} optimization problems are derived. A constraint qualification condition which uses the recent generalized second-order directional derivative is employed to obtain these conditions. Various second-order sufficient conditions are given under appropriate conditions on the generalized second-order directional derivative in a neighborhood of a given point. An application of the secondorder conditions to a new class of nonsmooth C^{1, 1} optimization problems with infinitely many constraints is presented.     

Multigrid solution of high order discretisation for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind

In this paper, we propose two compact finite difference approximations for three-dimensional biharmonic equation with Dirichlet boundary conditions of second kind. In these methods there is no need to define special formulas near the boundaries and boundary conditions are incorporated with these techniques. The unknown solution and its second derivatives are carried as unknowns at grid points. We derive second-order and fourth-order approximations on a 27 point compact stencil. Classical iteration methods such as Gauss–Seidel and SOR for solving the linear system arising from the second-order and fourth-order discretisation suffer from slow convergence. In order to overcome this problem we use multigrid method which exhibit grid-independent convergence and solve the linear system of equations in small amount of computer time. The fourth-order finite difference approximations are used to solve several test problems and produce high accurate numerical solutions.     

Generalized second-order directional derivatives and optimization with C1,1 functions

In this paper, a new generalized second-order directional derivative and a set-valued generalized Hessian are introudced for C¹,¹ functions in real Banach spaces. It is shown that this set-valued generalized Hessian is single-valued at a point if and only if the function is twice weakly Gãteaux differentiable at the point and that the generalized second-order directional derivative is upper semi-continuous under a regularity condition. Various generalized calculus rules are also given for C^1,1 functions. The generalized second-order directional derivative is applied to derive second-order necessary optirnality conditions for mathematical programming problems.     

The uv-Decomposition on a Class of D.C. Functions and Optimality Conditions

In this paper,the UV-theory and P-differential calculus are employed to study second-order ex-pansion of a class of D.C.functions and minimization problems.Under certain conditions,some properties ofthe U-Lagrangian,the second-order expansion of this class of functions along some trajectories are formulated.Some first and second order optimality conditions for the class of D.C.optimization problems are given.     

High-resolution large time-step schemes for inviscid fluid flow

We present a set of sufficient conditions for conservative, consistent and total variation diminishing (TVD) Large Time Step (LTS) schemes. We generalize the modified flux approach of Harten (1986) to achieve second-order accuracy away from discontinuities. In particular, we derive a supplementary condition, providing a full set of sufficient conditions for high-resolution LTS.In this framework, a second-order extension (LTS-Roe2) of the LTS-Roe scheme is proposed and compared to the second order LTS scheme proposed by Harten. Tests show that LTS-Harten consistently gives good results with less oscillations than LTS-Roe2, but has a tendency to smear out discontinuities when the Courant number is increased.     

Convergence of a third order method for fixed points in Banach spaces

In this paper, the semilocal convergence of a third order Stirling-like method used to find fixed points of nonlinear operator equations in Banach spaces is established under the assumption that the first Fréchet derivative of the involved operator satisfies ??-continuity condition. It turns out that this convergence condition is weaker than the Lipschitz and the H?lder continuity conditions on first Fréchet derivative of the involved operator. The importance of our work lies in the fact that numerical examples can be given to show that our approach is successful even in cases where Lipschitz and H?lder continuity conditions on first Fréchet derivative fail. It also avoids the evaluation of second order Fréchet derivative which is difficult to compute at times. A priori error bounds along with the domains of existence and uniqueness of a fixed point are derived. The R-order of the method is shown to be equal to (2p?+?1) for p????(0,1]. Finally, two numerical examples involving nonlinear integral equations are worked out to show the efficacy of our approach.     

Second-order risk comparison of SLSE with GLSE and MLE in a regression with serial correlation

First, the second-order bias of the estimator of the autoregressive parameter based on the ordinary least squares residuals in a linear model with serial correlation is given. Second, the second-order expansion of the risk matrix of a generalized least squares estimator with the above estimated parameter is obtained. This expansion is the same as that based on a suitable estimator of the autoregressive parameter independent of the sample. Third, it is shown that the risk matrix of the generalized least squares estimator is asymptotically equivalent to that of the maximum likelihood estimator up to the second order. Last, a sufficient condition is given for the term due to the estimation of the autoregressive parameter in this expansion to vanish under Grenander's condition for the explanatory variates.     

Second-order tests in optimization theories

There are two main kinds of second-order tests in optimization theories. The simplest tests (Refs. 1–13), such as the generalized Legendre-Clebsch condition, require a local study around points of the trajectory of interest (generally, on a singular arc). Tests of the second kind (Refs. 14–22 and 51–53) are more difficult to use but also more efficient: they can be applied under various assumptions of convexity or linearity and generally require some integrations along the trajectory of interest (usually, these integrations can only be done by numerical methods). In favorable cases, the better tests of this second kind lead to either the conclusion of nonoptimality or the conclusion of local optimality. This survey paper begins with a broader question, the question of sufficient conditions for absolute optimality. Some results of this study are used to define anadjoint matrix (extension of the notion ofadjoint vector of Pontryagin). Then, the different second-order tests can be unified and generalized even if the trajectory of interest has switches and singular arcs. On a given trajectory, theconjugate points are easily related to the evolution of the adjoint matrix. Finally, this generalized second-order test is applied to a singular arc of astrodynamics, the reversible arc: this arc is globally optimal from end to end.     

Solitary waves of the EW and RLW equations

Eight finite difference methods are employed to study the solitary waves of the equal-width (EW) and regularized long–wave (RLW) equations. The methods include second-order accurate (in space) implicit and linearly implicit techniques, a three-point, fourth-order accurate, compact operator algorithm, an exponential method based on the local integration of linear, second-order ordinary differential equations, and first- and second-order accurate temporal discretizations. It is shown that the compact operator method with a Crank–Nicolson discretization is more accurate than the other seven techniques as assessed for the three invariants of the EW and RLW equations and the L₂-norm errors when the exact solution is available. It is also shown that the use of Gaussian initial conditions may result in the formation of either positive or negative secondary solitary waves for the EW equation and the formation of positive solitary waves with or without oscillating tails for the RLW equation depending on the amplitude and width of the Gaussian initial conditions. In either case, it is shown that the creation of the secondary wave may be preceded by a steepening and an narrowing of the initial condition. The creation of a secondary wave is reported to also occur in the dissipative RLW equation, whereas the effects of dissipation in the EW equation are characterized by a decrease in amplitude, an increase of the width and a curving of the trajectory of the solitary wave. The collision and divergence of solitary waves of the EW and RLW equations are also considered in terms of the wave amplitude and the invariants of these equations.     

Monotone iterative method for the second-order three-point boundary value problem with upper and lower solutions in the reversed order

This article introduces a new concept of upper and lower solutions and studies the existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order. The sufficient conditions for the existence and uniqueness of solutions are obtained by using the monotone iterative method, meantime, the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution are given. Some results of previous literature are extended and improved. A numerical example is also included to illustrate the effectiveness of the proposed results.     

Mesh‐independent convergence of the modified inexact Newton method for a second order non‐linear problem

In this paper, we consider an inexact Newton method applied to a second order non‐linear problem with higher order non‐linearities. We provide conditions under which the method has a mesh‐independent rate of convergence. To do this, we are required, first, to set up the problem on a scale of Hilbert spaces and second, to devise a special iterative technique which converges in a higher than first order Sobolev norm. We show that the linear (Jacobian) system solved in Newton's method can be replaced with one iterative step provided that the initial non‐linear iterate is accurate enough. The closeness criteria can be taken independent of the mesh size. Finally, the results of numerical experiments are given to support the theory. Published in 2005 by John Wiley & Sons, Ltd.