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SIAM J. on Numerical Analysis

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1972

Volume 9, Issue 4, pp. 523-733


Convergent Finite Difference Schemes for Nonlinear Parabolic Equations

Albert C. Reynolds, Jr.

SIAM J. Numer. Anal. 9, pp. 523-533 (11 pages) | Cited 5 times

Online Publication Date: July 14, 2006

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In this paper, initial-boundary value problems for a general class of nonlinear parabolic equations are studied. We show the solutions of certain associated finite difference equations converge to the solution of the initial-boundary value problem with $O(h^2 )$ rate of convergence..

The Solution of Large Symmetric Eigenproblems by Sectioning

Paul S. Jensen

SIAM J. Numer. Anal. 9, pp. 534-545 (12 pages) | Cited 7 times

Online Publication Date: July 14, 2006

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When a relatively few eigenvalues are desired for a very large symmetric matrix eigenvalue problem, direct methods such as Householder reduction tend to be inefficient. Inverse iteration works reasonably well but runs into difficulties when eigenvalues are clustered. This paper presents a method for determining the eigenvalues lying in a “section” $\alpha < \lambda < \beta $ of the eigenvalue spectrum together with the corresponding eigenvectors. In contrast with inverse iteration, the sectioning method works particularly well for clustered eigenvalues.
The sectioning method proceeds in three phases : first, a basis for the invariant subspace corresponding to the spectral section $\alpha < \lambda < \beta $ is computed, next this basis is used to reduce the eigenproblem by the Ritz process, and finally, the reduced problem is solved in high precision by a fairly standard Householder technique.

Constructive Approximation of Solutions to Linear Elliptic Boundary Value Problems

N. L. Schryer

SIAM J. Numer. Anal. 9, pp. 546-572 (27 pages) | Cited 10 times

Online Publication Date: July 14, 2006

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Let $u$ be the solution of $Lu = \Delta u + au_x + bu_y + cu = 0$ on $D$, $u = f$ on $\partial D$, where $f$ is continuous, the coefficients $a,b$ and $c$ are analytic on $D$ and $c$ is nonpositive there. In Part I of this paper a set of particular solutions of $Lu = 0$ is constructed from the coefficients $a,b$ and $c$. These functions are independent of the domain $D$ and represent generalizations of the harmonic polynomials $\operatorname{Re} (x + iy)^k $ and $\operatorname{Im} (x + iy)^k $ for $\Delta u = 0$. Special attention is paid to domains with corners in their boundaries and another family of particular solutions of $Lu = 0$ is constructed with the proper asymptotic behavior at these corners. It is shown that the solution, $u$, of the boundary value problem for $Lu = 0$ may be approximated arbitrarily well using finite linear combinations of these particular solutions. The norm used is the maximum norm over the closure of the domain $D$.
Part II shows how to obtain such approximating linear combinations in practice. The technique is quite simple: choose $M$ points on $\partial D$ and find a linear combination of $N$ of these particular solutions, $N \leqq M$, which best approximates the given boundary values $f$ at these $M$ points. Obtaining such a linear combination is a linear programming problem and may be solved in practice. It is shown that as $N$ and $M$ go to infinity the computed linear combinations approach the solution of the problem uniformly.
A posteriors error bounds are given for these approximate solutions of the boundary value problem. Finally, numerical results are presented for the special case where $a,b$ and $c$ are polynomials in $x$and $y$.

A Unified Approach to Quadrature Rules with Asymptotic Estimates of Their Remainders

J. D. Donaldson and David Elliott

SIAM J. Numer. Anal. 9, pp. 573-602 (30 pages) | Cited 24 times

Online Publication Date: July 14, 2006

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The starting point of this paper is a theorem, based on the theory of functions of a complex variable, which essentially gives a representation of the remainder in a quadrature rule as a contour integral. From this theorem it is shown how most of the well-known interpolatory and non-interpolatory quadrature rules may be derived. Some new quadrature rules are also given. The latter part of the paper is devoted to the evaluation of asymptotic estimates of the remainder, by consideration of particular examples. It is shown how this asymptotic analysis frequently gives excellent estimates of the remainder.

Comparing Numerical Methods for Ordinary Differential Equations

T. E. Hull, W. H. Enright, B. M. Fellen, and A. E. Sedgwick

SIAM J. Numer. Anal. 9, pp. 603-637 (35 pages) | Cited 66 times

Online Publication Date: July 14, 2006

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Numerical methods for systems of first order ordinary differential equations are tested on a variety of initial value problems. The methods are compared primarily as to how well they can handle relatively routine integration steps under a variety of accuracy requirements, rather than how well they handle difficulties caused by discontinuities, stiffness, roundoff or getting started.
According to criteria involving the number of function evaluations, overhead cost, and reliability, the best general-purpose method, if function evaluations are not very costly, is one due to Bulirsch and Stoer. However, when function evaluations are relatively expensive, variable-order methods based on Adams formulas are best. The overhead costs are lower for the method of Bulirsch and Stoer, but the Adams methods require considerably fewer function evaluations. Krogh’s implementation of a variable-order Adams method is the best of those tested, but one due to Gear is also very good. In general, Runge–Kutta methods are not competitive, but fourth or fifth order methods of this type are best for restricted classes of problems in which function evaluations are not very expensive and accuracy requirements are not very stringent.
The problems, methods and comparison criteria are specified very carefully. One objective in doing so is to provide a rigorous conceptual basis for comparing methods. Another is to provide a useful standard for such comparisons. A program called DETEST is used to obtain the relevant statistics for any particular method.

Numerical Chebyshev Approximation in the Complex Plane

Jack Williams

SIAM J. Numer. Anal. 9, pp. 638-649 (12 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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The paper describes a numerical method for computing rational approximations of analytic functions on arbitrary simply-connected closed regions of the complex plane. The method is based on approximation on finite point sets and can be regarded as a generalization of the exchange algorithm for real rational approximation. Numerical examples show that for practical purposes, the approximations may be regarded as optimal in the sense of Chebyshev.

Strang-Type Difference Schemes for Multidimensional Problems

David Gottlieb

SIAM J. Numer. Anal. 9, pp. 650-661 (12 pages) | Cited 6 times

Online Publication Date: July 14, 2006

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Using Strang’s idea, explicit difference schemes of second order accuracy and of optimal stability are obtained for solving partial differential equations in $d$ dimensions.

A Simple Interpolation Algorithm for Improvement of the Numerical Solution of a Differential Equation

Bengt Lindberg

SIAM J. Numer. Anal. 9, pp. 662-668 (7 pages) | Cited 1 time

Online Publication Date: July 14, 2006

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Extrapolation methods for the numerical solution of systems of differential equations give very accurate solutions on a coarse grid. An algorithm for computing accurate solutions for points not on that grid is developed. The algorithm is based on interpolation of certain functions which are constructed recursively from the successive extrapolated values. Some numerical results are presented which show the advantages of the method.

On the Sensitivity of the Eigenvalue Problem $Ax = \lambda Bx$

G. W. Stewart

SIAM J. Numer. Anal. 9, pp. 669-686 (18 pages) | Cited 39 times

Online Publication Date: July 14, 2006

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This paper considers the sensitivity of the eigenvalues and eigenvectors of the generalized matrix eigenvalue problem $Ax = \lambda Bx$ to perturbations of $A$ and $B$. The notion of a deflating subspace for the problem is introduced, and error bounds for approximate deflating subspaces obtained. The bounds also provide information about the eigenvalues of the problem. The resulting perturbation bounds estimate realistically the sensitivity of the eigenvalues, even when $B$ is singular or nearly singular. The results are applied to the important special case where $A$ is Hermitian and $B$ is positive definite.

Local Convergence of Smooth Cubic Spline Interpolates

W. J. Kammerer and G. W. Reddien, Jr.

SIAM J. Numer. Anal. 9, pp. 687-694 (8 pages) | Cited 2 times

Online Publication Date: July 14, 2006

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In this paper we develop local error bounds for smooth cubic spline interpolates of a function $f$ which depend only on the local smoothness of $f$. Moreover, the rate of convergence will be the same as if $f$ possessed this degree of smoothness over the entire interval.

Stable Matching of Difference Schemes

Melvyn Ciment

SIAM J. Numer. Anal. 9, pp. 695-701 (7 pages) | Cited 3 times

Online Publication Date: July 14, 2006

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Approximations that result from the natural matching of two stable dissipative difference schemes across a coordinate line are shown to be stable. The basic idea is to reformulate the matching scheme consistent to an equivalent initial boundary value problem and to verify the algebraic conditions for stability of such systems. An interesting comparison to the above result is the case of redefinition of a scheme at a single point. In particular, we show that some unstable perturbations do not upset the stability of the Lax–Wendroff scheme.

Galerkin Methods for Vibration Problems in Two Space Variables

Graeme Fairweather

SIAM J. Numer. Anal. 9, pp. 702-714 (13 pages) | Cited 5 times

Online Publication Date: July 14, 2006

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Some discrete Galerkin methods are formulated for approximately solving a linear fourth order parabolic partial differential equation in $\Omega \times (0,T]$, where $\Omega $ is an arbitrary bounded domain in $\mathbb{R}^2 $. Error estimates are obtained which show that these methods are second order correct in time. When $\Omega $ is a rectangle, alternating-direction Galerkin methods are discussed.

Multistep Methods with Variable Matrix Coefficients

J. D. Lambert and S. T. Sigurdsson

SIAM J. Numer. Anal. 9, pp. 715-733 (19 pages) | Cited 14 times

Online Publication Date: July 14, 2006

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In a previous paper concerning linear multistep methods with mildly varying coefficients for the numerical solution of a single ordinary differential equation, a certain stabilizing condition was derived. Satisfaction of this condition ensured, under certain hypotheses, that the parasitic solutions of the difference equation which arise when such a method is applied to a nonlinear differential equation would not dominate the genuine solution. In this paper, it is shown that the application of a generalization of this stabilizing condition to a somewhat wider class of linear multistep methods—which are applicable to systems of differential equations—results in a class of methods with variable matrix coefficients. It is shown that methods of this class, with suitably chosen coefficients, possess a stability property similar to A-stability, and the asymptotic stability behavior of the methods when applied to a variable coefficient linear system is also investigated. Explicit and implicit examples of the new methods of order up to four are quoted, and numerical tests are reported.
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