Optimum Extrapolated Alternating Direction Implicit Schemes in the Presence of Singular Matrices

This paper shows how Extrapolated Alternating Direction Implicit(E.A.D.I.) methods can be used for the numerical solution ofLaplace's equation under Neumann boundary conditions. E.A.D.I.methods are applied with the Douglas set of parameters and optimumE.A.D.I. schemes are given.     

The numerical solution of a model problem biharmonic equation by using Extrapolated Alternating Direction Implicit methods

The numerical solution of the 2-dimensional biharmonic equation over the unit square by using Extrapolated Alternating Direction Implicit (E.A.D.I.) methods is studied. To approximate the biharmonic equation both a 13-point and a 25-point difference replacements are considered. In each case E.A.D.I. schemes are used together with the acceleration parameter fixed during the iterations or varying according to the Douglas set of parameters. Finally optimum E.A.D.I. schemes are given for every value of the numberN of mesh subdivisions in each co-ordinate direction.     

Numerical Solution of the Robin Problem by Extrapolated A. D. I. Methods

In this paper the Robin problem for elliptic equations is consideredand its numerical integration by several variants of the Extrapolatedform of the Alternating Direction Implicit (E.A.D.I.) methodsis discussed. A comparison among the E.A.D.I. variants includedhere as well as those studied in an earlier paper is carriedout.     

Three-level extrapolated alternating direction implicit (E.A.D.I.) methods for the numerical solution of two-dimensional second order elliptic problems

This paper extends the theory concerning the three-level E.A.D.I. schemes to cover the numerical solution of more general (than the trivial first boundary value problem for Poisson's equation in a square) two-dimensional second order elliptic problems. Moreover numerical examples proving the validity of the theory developed are presented and general conclusions are drawn.     

On Chebychev Acceleration Procedures for Alternating Direction Iterative Methods

The solution of large sparse systems of linear equations arising,for example, from the numerical solution of elliptic partialdifferential equations is considered, with reference to theacceleration technique commonly known as Chebychev acceleration.In particular its application to alternating direction iterative(A.D.I.) methods is compared with the more standard techniquessuch as successive overrelaxation. It is conjectured that inmost circumstances a suitable A.D.I. strategy is that of applyingChebychev semiiteration to an A.D.I. process with a single A.D.I.parameter. It is shown that under general conditions this procedure maysometimes produce faster convergence than the usual multiparameterA.D.I. procedure.     

High accuracy A.D.I. methods for fourth order parabolic equations with variable coefficients

High accuracy alternating direction implicit (A.D.I.) methods are derived for solving fourth order parabolic equations with variable coefficients in one, two, and three space dimensions. Splittings are discussed and numerical results are presented.     

Numerical experiments with a multistep Radau method

This paper describes an implementation of multistep collocation methods, which are applicable to stiff differential problems, singular perturbation problems, and D.A.E.s of index 1 and 2.These methods generalize one-step implicit Runge-Kutta methods as well as multistep one-stage BDF methods. We give numerical comparisons of our code with two representative codes for these methods, RADAU5 and LSODE.     

Connections between projective geometry and superassociative algebra

This paper establishes a connection between projective geometry and the superassociative algebra of multiplace functions. In Part I a quaternary operation is defined for the points on a line j in a projective plane π relative to a fixed quadrangle and the result of operating on A,B,C,D is denoted by A(B,C,D). It is shown that π is a Pappus plane if and only if this operation is superassociative: (A(B,C,D))(E,F,G)=A(B(E,F,G),C(E,F,G),D(E,F,G)) for any points A,B,C,D,E,F,G on j. Desargues and certain special Desargues planes are also characterized by restrictions of the superassociative law. In Part II projective geometry is developed over superassociative systems.     

Numerical analysis for factorial data analysis. Part I: Numerical software — the package INDA for microcomputers

A large collection of factorial data analysis methods can be characterized by the following matrices: X , the k x n matrix of data, and A, B the symmetric positive definite matrices of size n, k which represent the chosen norms of ?ⁿ, ?^k, respectively. All methods amount to computing the largest eigenvalues of U = XAX^TB or the largest singular values of E = B^1/2XA^1/2 . In Part I of this paper we begin by a geometrical and probabilistic interpretation of the various methods, showing how U and E are defined in each case. We then define the computational kernel for factorial data analysis. We conclude by devising the numerical aspects of software implementation for this kernel on microcomputers and presenting the package INDA.     

Convergence of a fixed point iteration method for the OSV model

In image processing, image denoising and texture extraction are important problems in which many new methods recently have been developed. One of the most important models is the OSV model [S. Osher, A. Solé, L. Vese, Image decomposition and restoration using total variation minimization and the H^-1 norm, Multiscale Model. Simul. A SIAM Interdisciplinary J. 1(3) (2003) 349-370] which is constructed by the total variation and H^-1 norm. This paper proves the existence of the minimizer of the functional from the OSV model and analyzes the convergence of an iterative method for solving the problems. Our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated Euler-Lagrange equation, and the limit of our iterations satisfies the minimizer of the functional from the OSV model. In numerical experiments, we compare the numerical results of our works with those of the ROF model [L.I. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D 60 (1992) 259-268].     

High order methods for the numerical solution of two-point boundary value problems

In a recent paper, Cash and Moore have given a fourth order formula for the approximate numerical integration of two-point boundary value problems in O.D.E.s. The formula presented was in effect a one-off formula in that it was obtained using a trial and error approach. The purpose of the present paper is to describe a unified approach to the derivation of high order formulae for the numerical integration of two-point boundary value problems. It is shown that the formula derived by Cash and Moore fits naturally into this framework and some new formulae of orders 4, 6 and 8 are derived using this approach. A numerical comparison with certain existing finite difference methods is made and this comparison indicates the efficiency of the high order methods for problems having a suitably smooth solution.     

Numerical Solution of the Third Boundary Value Problem for Laplace's Equation

This paper describes how a numerical solution of Laplace's equationunder Robbins boundary conditions over a unit square may bereached by using an A.D.I. scheme with various accelerationparameters, the determination of which is outlined, and comparisonsamong which are made.     

Dispersive properties of a viscous numerical scheme for the Schrödinger equation

In this Note we study the dispersive properties of the numerical approximation schemes for the free Schrödinger equation. We consider finite-difference space semi-discretizations. We first show that the standard conservative scheme does not reproduce at the discrete level the properties of the continuous Schrödinger equation. This is due to spurious high frequency numerical solutions. In order to damp out these high-frequencies and to reflect the properties of the continuous problem we add a suitable extra numerical viscosity term at a convenient scale. We prove that the dispersive properties of this viscous scheme are uniform when the mesh-size tends to zero. Finally we prove the convergence of this viscous numerical scheme for a class of nonlinear Schrödinger equations with nonlinearities that may not be handeled by standard energy methods and that require the so-called Strichartz inequalities. To cite this article: L.I. Ignat, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

High Accuracy A.D.I. Methods for Hyperbolic Equations with Variable Coefficients

High accuracy alternating direction implicit (A.D.I.) methodsare derived for solving the wave equation with variable coefficientsin one, two, and three space dimensions. Splittings are discussedand numerical results presented.     

矩阵不等式约束下矩阵方程AX=B的双对称解

本文讨论矩阵不等式CXD≥E 约束下矩阵方程AX=B的双对称解,即给定矩阵A,B,C,D和 E, 求双对称矩阵X, 使得AX=B 和 CXD≥E, 其中CXD≥E表示矩阵CXD-E非负.本文将问题转化为矩阵不等式最小非负偏差问题,利用极分解理论给出了求其解的迭代方法,并结合相关矩阵理论说明算法的收敛性.最后给出数值算例验证算法的有效性.     

High order approximations using spline-based differential quadrature method: Implementation to the multi-dimensional PDEs

A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.     

A modified variable time step method for solving ice melting problem

A modified numerical method was used by authors for solving 1D Stefan problem. In this paper a modified method is proposed with difference formulae and different methods of calculating the variable time step, which are deduced from Taylor series expansions of different conditions at the boundary. Also an extrapolation formula for the solution at the first point at the right of the computational domain is proposed. The numerical results are compared with those obtained from other methods.     

On the stability of alternating‐direction explicit methods for advection‐diffusion equations

Alternating‐Direction Explicit (A.D.E.) finite‐difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well‐known, as are stable A.D.E. schemes for solving the first‐order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time‐dependent advection‐diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi‐linear one‐dimensional advection‐diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Extrapolated alternating direction implicit iterative methods

This paper describes a new generalised (extrapolated) A.D.I. method for the solution of Laplace's equation. This method uses (i) a fixed acceleration parameter and (ii) the set of acceleration parameters of Douglas. The theory is applied to the 2-dimensional case and optimum numerical results are obtained.     

Error estimate for the numerical solution of Fredholm's integral equation

A bound on the actual error incurred in the numerical solution of Fredholm's integral equations with symmetric kernel is obtained, using computed approximations of some eigenvalues.This article is a part of D. Sc. Thesis of E. Rakotch.