A High Accuracy Alternating Direction Method for the Wave Equation

The alternating direction implicit (ADI) method of Lees forsolving the wave equation in two space dimensions is generalizedand an ADI method of increased accuracy obtained. The new methodis demonstrated by a numerical example. An extension is alsogiven to cover the case of the wave equation in three spacedimensions.     

A fourth-order cubic spline method for linear second-order two-point boundary-value problems

A cubic spline method for linear second-order two-point boundary-valueproblems is analysed. The method is a Petrov-Galerkin methodusing a cubic spline trial space, a piecewise-linear test space,and a simple quadrature rule for the integration, and may beconsidered a discrete version of the H¹-Galerkin method. Themethod is fully discrete, allows an arbitrary mesh, yields alinear system with bandwidth five, and under suitable conditionsis shown to have an 0(h^4– rate of convergence in the W_p¹norm for i = 0, 1, 2, 1p. The H¹-Galerkin method and orthogonalspline collocation with Hermite cubics are also discussed.     

On the Application of Extrapolation, Deferred Correction and Defect Correction to Discrete-time Galerkin Methods for Parabolic Problems

Conditions which ensure the applicability of extrapolation-in-timeof the backward-difference Galerkin and Crank—NicolsonGalerkin methods for linear parabolic initial-boundary problemsare discussed. Deferred correction and defect correction proceduresfor these discrete-time Galerkin methods are formulated andtheir theoretical properties summarized. These procedures yieldapproximations which are up to fourth-order correct in time,and whose computation requires less work per time step thanthat required by extrapolation. The results of numerical experiments,which demonstrate the expected rates of convergence with respectto the time discretization of the various techniques, are presented. ^*The work of this author was supported in part by the NationalScience Foundation under grant MCS-8102295     

A Box Method for a Nonlinear Equation of Population Dynamics

We analyse a box method for a nonlinear integro-differentialequation with a nonlocal boundary condition. The problem considereddescribes the evolution in time of the age structure of a population.We study the consistency, stability, and convergence propertiesof the scheme and show that it is second-order accurate. Thisanalysis is carried out employing an abstract theory of discretizations.     

The Method of Fundamental Solutions for the Solution of Nonlinear Plane Potential Problems

The method of fundamental solutions is described for the solutionof elliptic boundary value problems governed by Laplace's equationin the plane subject to nonlinear radiation-type boundary conditions.The effectiveness of the method is demonstrated by examiningits performance on two problems from the literature, and comparisonsare made with published results obtained using boundary elementmethods. Portions of this work were conducted while this author wasa visiting assistant professor at the University of Kentucky. Partially supported by the National Science Foundation undergrant MCS-8303287 and grant RII-8610671, and by the Commonwealthof Kentucky through the Kentucky EPSCoR Program.     

On the Approximate Solution of a Diffusion Problem by Galerkin Methods

Discrete-time Galerkin methods are considered for the approximatesolution of a parabolic initial boundary value problem whicharises, for example, in problems involving the diffusion ofa solute into a solid from a stirred solution of fixed volume.Optimal error estimates in the L² and H¹ norms are derived forthe Crank-Nicolson Galerkin method. For the one space variablecase optimal L estimates are also obtained. Results of numericalexperiments are presented and comparisons with finite differenceapproximations are made.