A Mushy Region in a Stefan Problem

In the melting of a pure metal by volume heating, when superheatingdoes not occur, there will be a region in which the metal isneither pure liquid nor pure solid, called a mushy region. Agrain or dendrite model is proposed to describe the microstructureof the metal in this situation, and its stability is discussed.The explicit solution for the one-dimensional version of themodel is obtained and has a different form in three regions;near the pure solid boundary, in the mush, and near the pureliquid boundary. An appropriate average of this solution forthe microstructure is shown to reduce to the weak solution forthe microstructure is shown to reduce to the weak solution forthe macroscopic problem proposed by Atthey (1974). Propertiesof the solution for more general grain geometries are also discussed.     

On the limiting profile arising from orthonormalizing shifts of exponentially decaying functions

It is shown that, under certain conditions, orthonormalizingthe positive integer shifts of an exponentially decaying functionon the half line by the Gram-Schmidt process leads to a limitingprofile given by orthonormalizing all their integer shifts onthe whole line. These results derive from properties of Choleskyfactorization of bi-infinite and semi-infinite matrices. Anexample is provided by the negative exponential function andconjectures are given, supported by numerical evidence, forthe Gaussian and Lorentz function.     

关于隐式代数曲面的一种几何连续性定义

本文利用函数的幂级数给出了隐式代数曲面的一种几何连续性定义．证明了在此定义下的ｋ阶几何连续即为ｋ阶变尺度（ｒｅｓｃａｌｉｎｇ）连续，并举例说明了其应用的方便性．     

Error bounds for the solution to the algebraic equations in Runge-Kutta methods

In the implementation of implicit Runge-Kutta methods inaccuracies are introduced due to the solution of the implicit equations. It is shown that these errors can be bounded independently of the stiffness of the differential equation considered if a certain condition is satisfied. This condition is also sufficient for the existence and uniqueness of a solution to the algebraic equations. TheBSI-andBS-stability properties of several classes of implicit methods are established.     

Implicit difference methods for Hamilton-Jacobi functional differential equations

Classical solutions of initial boundary value problems are approximated by solutions of associated implicit difference functional equations. A stability result is proved by using a comparison technique with nonlinear estimates of the Perron type for given functions. The Newton method is used to numerically solve nonlinear equations generated by implicit difference schemes. It is shown that there are implicit difference schemes which are convergent whereas the corresponding explicit difference methods are not. The results obtained can be applied to differential integral problems and differential equations with deviated variables.     

求解一类隐线性互补问题的投影MAOR算法

用MAOR迭代算法求解一类L-矩阵的隐线性互补问题.证明了由此算法产生的迭代序列的聚点是隐线性互补问题的解.并且当问题中的矩阵是M-矩阵时,算法产生的迭代序列单调收敛于隐互补问题的解.     

Linear quadratic control problem with a terminal convex constraint for discrete-time distributed systems

The present work deals with the linear quadratic control problemfor a discrete distributed system with terminal convex constraint.Using techniques of perturbation by feedback, it is shown thatthe resolution of the considered problem is equivalent to thatof a controllability, one so-called Extended Exact Controllabilitywith time-varying operators. The Hilbert uniqueness method approachis then extended to this case to provide an explicit form forthe optimal control. In the same framework, the inequality constraintcase is examined for which a practical numerical resolutionis given. Finally, the results obtained are used to treat aminimum-time reachability problem.     

Implicit ad hoc methods for nonlinear partial differential equations

Several special methods including implicit separation of variables, explicit and implicit generalized traveling waves are introduced and employed to obtain solutions for nonlinear equations. Certain nonlinear wave propagation problems are shown to yield to implicit separation while generalized traveling wave concepts are applied in diffusion, fluid mechanics and wave propagation.     

Numerical methods for Hamilton Jacobi functional differential equations

Initial and initial boundary value problems for first order partial functional differential equations are considered. Explicit difference schemes of the Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are construted.     

Non-smooth data error estimates for linearly implicit Runge-Kutta methods

Linearly implicit time discretizations of semilinear parabolicequations with non-smooth initial data are studied. The analysisuses the framework of analytic semigroups which includes reaction-diffusionequations and the incompressible Navier-Stokes equations. Itis shown that the order of convergence on finite time intervalsis essentially one. Applications to the long-term behaviourof linearly implicit Runge-Kutta methods are given.     

Fixed-Point Technique for Implicit Complementarity Problem in Hilbert Lattice

In this paper, we consider and study the implicit complementarity problem in the setting of a Hilbert lattice. It has been shown that this problem can be formulated as a fixed-point problem by using a suitable change of variables. Moreover, this formulation allows us to prove the existence and uniqueness of solutions of the implicit complementarity problem.     

Vector-valued Rational Interpolants II

Formulae for rational interpolation of vector data (in a spaceC^[d]) at distinct points are given. Its confluent case of vector-valuedPadé approximation is shown to be equivalent to the Germanpolynomial approximation problem. Formulae are given for thevector of numerator polynomials and for the denominator polynomial.A continued fraction interpolant for vector data is also given.The methods are characterized by their requirement that certaindistinguished directions in the space C^[d] form part of thespecification. The case of matrix Padé approximants forthe partial realization problem is explicitly discussed.     

Expected Conditioning

A perturbation analysis based on probabilistic arguments isdeveloped for a range of problems in numerical linear algebra,including well-determined and over-determined linear systems.Condition matrices and condition numbers are determined forthe expected value of the actual condition number of a problem.These enable attainable lower and upper bounds on the expectedcondition properties of a matrix to be given, independent ofany particular linear system. These estimates are much morereliable than those derived from conventional norm conditionnumbers, and are shown to reveal features which the latter cannot. The expected condition analysis has desirable properties underscaling transformations which is not the case for the norm conditionanalysis. It is shown that an optimal (or natural) scaling canbe associated with any matrix which moreover is readily computed.This enables the equilibration of a matrix to be carried out.Because this process is a uniquely defined projection it alwaysenables the best conditioned of all the possible equilibratedmatrices to be determined.     

Inexact implicit methods for monotone general variational inequalities

Solving a variational inequality problem is equivalent to finding a solution of a system of nonsmooth equations. Recently, we proposed an implicit method, which solves monotone variational inequality problem via solving a series of systems of nonlinear smooth (whenever the operator is smooth) equations. It can exploit the facilities of the classical Newton–like methods for smooth equations. In this paper, we extend the method to solve a class of general variational inequality problems Moreover, we improve the implicit method to allow inexact solutions of the systems of nonlinear equations at each iteration. The method is shown to preserve the same convergence properties as the original implicit method. Received July 31, 1995 / Revised version received January 15, 1999? Published online May 28, 1999     

A stability property of implicit Runge-Kutta methods

A class of implicit Runge-Kutta methods is shown to possess a stability property which is a natural extension of the notion ofA-stability for non-linear systems.     

Update Procedures and the 1‐Consistency of Arithmetic

The 1‐consistency of arithmetic is shown to be equiva ent to the existence of fixed points of a certain type of update procedure, which is implicit in the epsilon‐substitution method.     

二维抛物型方程的高精度多重网格解法

提出了数值求解二维抛物型方程的一种新的高精度加权平均紧隐格式，利用Fourier分析方法证明了该格式是无条件稳定的，为了克服传统迭代法在求解隐格式是收敛速度慢的缺陷，利用了多重网格加速技术，大大加快了迭代收敛速度，提高了求解效率，数值实验结果验证了方法的精确性和可靠性。     

Stable finite-element calculation of incompressible flows using the rotation form of convection

Conforming finite-element approximations are considered forthe incompressible Navier–Stokes equations with nonlinearterms written in the convection or rotation forms. Implicittime integration results in nice stability properties of auxiliaryproblems which can be solved by efficient numerical algorithms.The original nonlinear system admits relatively simple stabilizationstrategies. The paper presents in a unified form the convergenceanalysis, including the design of stabilization parameters,for linearized equations in both convection and rotation forms.Moreover, it is shown that a Galerkin discretization of thepressure-regularized Oseen problem with skew-symmetric termsin rotation form possesses better stability properties and,being much easier to solve, can be used as a predictor in implicitcalculations.     

Stable non-standard implicit finite difference schemes for non-linear heat transfer in a thin finite rod

In this study, first, three non-standard implicit finite difference schemes are proposed for solving the initial-boundary value problem involving a quartic non-linearity that arises in heat transfer involving conduction with thermal radiation. A thin finite rod exposed to radiating heat across its lateral surface into a medium of constant temperature and convection is ignored. Stability and consistency of the third scheme is proved. Numerical results are compared with non-standard explicit finite difference schemes that show fully stability of our third proposed scheme. Then, three non-standard implicit and three non-standard explicit finite difference schemes are proposed for solving the heat transfer problem with additional convection term. It is shown that in the second case when the model involves conduction, radiation and convection terms, the rod reaches steady state sooner. Numerical results for implicit and explicit schemes are compared and the effect of the convection term is discussed.     

On projected Runge-Kutta methods for differential-algebraic equations

Ascher and Petzold recently introducedprojected Runge-Kutta methods for the numerical solution of semi-explicit differential-algebraic systems of index 2. Here it is shown that such a method can be regarded as the limiting case of a standard application of a Runge-Kutta method with a very small implicit Euler step added to it. This interpretation allows a direct derivation of the order conditions and of superconvergence results for the projected methods from known results for standard Runge-Kutta methods for index-2 differential-algebraic systems, and an extension to linearly implicit differential-algebraic systems.