S. McKee

The construction of hopscotch methods for parabolic and elliptic equations in two space dimensions with a mixed derivative

Hopscotch, a fast finite difference technique, is used to solve parabolic and elliptic equations in two space dimensions with a mixed derivative. The method is compared numerically with existing alternating direction implicit (A.D.I.) and locally one dimensional (L.O.D.) methods for simple problems. Douglas and Gunns A.D.I. method is both simplified and improved by reformulating it as a hopscotch method.

Product integration methods for second-kind Abel integral equations

The construction and convergence of high-order product integration methods for the second-kind Abel equation are discussed and the results of De Hoog and Weiss are generalised. Backward difference methods are introduced, and numerical results are presented which verify the theoretical rates of convergence.

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

Abstract 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.

Best convergence rates of linear multistep methods for Volterra first kind equations

In two papers Holyhead et al. (1975, 1976) analyzed the convergence of general linear multistep methods under minimum continuity assumptions. This paper is concerned with determining the maximum orders of convergence of these methods given that the truncation error has an asymptotic expansion with sufficiently many terms.ZusammenfassungDie Konvergenzeigenschaften von linearen Mehrschrittverfahren für Volterrasche Integralgleichungen erster Art mit minimalen Stetigkeitsbedingungen wurden in zwei Arbeiten von Holyhead et al. (1975, 1976) analysiert. Die vorliegende Arbeit beschäftigt sich, unter der Voraussetzung, daß der Diskretisierungsfehler eine hinreichende asymptotische Entwicklung besitzt, mit der Frage nach der maximalen Konvergenzordnung.

Hopscotch methods for elliptic partial differential equations

Abstract This paper analyses hopscotch algorithms when used to solve elliptic partial differential equations. A comparison with standard methods is made for the model problem.

Convergence analysis of discretization methods for nonlinear first kind volterra integral equations

SummaryAn existence and uniqueness result is given for nonlinear Volterra integral equations of the first kind. This permits, by means of analogous discrete manipulations, a general convergence analysis for a wide class of discretization methods for nonlinear first kind Volterra integral equations to be presented. A concept of optimal consistency allows twosided error bounds to be derived.

Convergence of linear multistep methods for volterra first kind equations with k(t,t)≡0

This paper is concerned with Volterra integral equations of the first kind whose kernel,k(t,s), is identically zero whent=s. The concepts of zero-stability and weak zero-stability are introduced and convergence results under the assumption that the truncation error has an asymptotic expansion with a certain number of terms are presented. Simple numerical examples verifying these rates of convergence are given.ZusammenfassungDiese Arbeit befaßt sich mit Volterraschen Integralgleichungen erster Art, deren Kernk(t,s) fürt=s identisch verschwindet. Die Begriffe Null-Stabilität und schwache Null-Stabilität werden eingeführt und Konvergenzresultate unter der Voraussetzung angegeben, daß der Verfahrensfehler eine asymptotische Entwicklung mit einer gewissen Anzahl von Gliedern besitzt. Weiters werden einfache numerische Beispiele angegeben, die diese Konvergenzraten bestätigen.