Addendum to “Stronger than uniform convergence of multistep difference methods”

This paper fills a gap which remained in Stronger than Uniform Convergence of Multistep Difference Methods, Numerische Mathematik 8, 29 (1966), namely to show that from the stability of the homogeneous difference operator on the particular basis chosen in the solution space one can infer its stability on the whole solution space.     

On the convergence of advanced linear multistep methods

A convergence theorem is given showing that zero-stable advanced linear multistep methods with orderp consistency have orderp convergence.     

On the convergence of multistep methods for nonlinear stiff differential equations

Summary Convergence estimates are given forA()-stable multistep methods applied to singularly perturbed differential equations and nonlinear parabolic problems. The approach taken here combines perturbation arguments with frequency domain techniques.     

Two-sided stability and convergence of multistage and multistep methods for ordinary differential equations

A norm is introduced which allows the extension of bistability and biconvergence results of Stummel (“Topics in Numerical Analysis, II,” Academic Press, New York, 1975; “Approximation Methods in Analysis,” Aarhus Universiteit, 1973) (which apply to one-step methods) to the case of multistage and multistep methods.     

Almost uniform convergence

The almost uniform convergence is between uniform and quasi-uniform one. We give some necessary and sufficient conditions under which the almost uniform convergence coincides on compact sets with uniform, quasi-uniform or uniform convergence, respectively. In the second section continuity of almost uniform limits is considered. Finally we characterize the set of all points at which a net of functions is almost uniformly convergent to a given function.     

Block‐centered upwind multistep difference method and convergence analysis for numerical simulation of oil reservoir

The three‐dimensional displacement of two‐phase flow in porous media is a preliminary problem of numerical simulation of energy science and mathematics. The mathematical model is formulated by a nonlinear system of partial differential equations to describe incompressible miscible case. The pressure is defined by an elliptic equation, and the concentration is defined by a convection‐dominated diffusion equation. The pressure generates Darcy velocity and controls the dynamic change of concentration. We adopt a conservative block‐centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved one order. We use a block‐centered upwind multistep method to solve the concentration, where the time derivative is approximated by multistep method, and the diffusion term and convection term are treated by a block‐centered scheme and an upwind scheme, respectively. The composite algorithm is effective to solve such a convection‐dominated problem, since numerical oscillation and dispersion are avoided and computational accuracy is improved. Block‐centered method is conservative, and the concentration and the adjoint function are computed simultaneously. This physical nature is important in numerical simulation of seepage fluid. Using the convergence theory and techniques of priori estimates, we derive optimal estimate error. Numerical experiments and data show the support and consistency of theoretical result. The argument in the present paper shows a powerful tool to solve the well‐known model problem.     

Multirate linear multistep methods

The design of a code which uses different stepsizes for different components of a system of ordinary differential equations is discussed. Methods are suggested which achieve moderate efficiency for problems having some components with a much slower rate of variation than others. Techniques for estimating errors in the different components are analyzed and applied to automatic stepsize and order control. Difficulties, absent from non-multirate methods, arise in the automatic selection of stepsizes, leading to a suggested organization of the code that is counter-intuitive. An experimental code and some initial experiments are described.Dedicated to Professor Germund Dahlquist on the occasion of his 60th birthdaySupported in part by the Department of Energy under grant DOE DEAC0276ERO2383.Work done while attending the University of Illinois.     

Symmetric linear multistep methods

Some important early contributions of Germund Dahlquist are reviewed and their impact to recent developments in the numerical solution of ordinary differential equations is shown. This work is an elaboration of a talk presented in the Dahlquist session at the SciCADE05 conference in Nagoya. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 65L06, 65P10     

On uniform convergence of measures

Summary A new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsØe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.Most of the work on the paper was done while the author was with the Institute for Information Theory and Automation of Czechoslovak Academy of Sciences. After August 21, 1968 the author worked on it while a guest of the Forschungsinstitut für Mathematik, Eidgenössische Technische Hochschule, Zürich, and the paper was completed at Michigan State University.     

On uniform convergence of approximation methods for operator equations of the second kind

Schock (1985) has considered the convergence properties of various Galerkin-like methods for the approximate solution of the operator equation of the second kind x - Tx = y, where T is a bounded linear operator on a Banach space X, and x and y belong to X, and proved that the classical Galerkin method and in certain cases, the iterated Galerkin method are arbitrarily slowly convergent whereas the Kantororich method studied by him is uniformly convergent. It is the purpose of this paper to introduce a general class of approximations methods for x - Tx = y which includes the well-known methods of projection and the quadrature methods, and to characterize its uniform convergence, so that an arbitrarily slowly convergent method can be modified to obtain a uniformly convergent method.     

Exponential multistep methods of Adams-type

The paper is concerned with the construction, implementation and numerical analysis of exponential multistep methods. These methods are related to explicit Adams methods but, in contrast to the latter, make direct use of the exponential and related matrix functions of a (possibly rough) linearization of the vector field. This feature enables them to integrate stiff problems explicitly in time.     

Global and uniform convergence of subspace correction methods for some convex optimization problems

This paper gives some global and uniform convergence estimates for a class of subspace correction (based on space decomposition) iterative methods applied to some unconstrained convex optimization problems. Some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems.

     

About the statistical uniform convergence

In this work we study the concept of statistical uniform convergence. We generalize some results of uniform convergence in double sequences to the case of statistical convergence. We also prove a basic matrix theorem with statistical convergence.     

On the uniform convergence of a finite difference scheme for time dependent singularly perturbed reaction-diffusion problems

In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction-diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method.