A highorder difference method for differential equations
Authors:
Robert E. Lynch and John R. Rice
Journal:
Math. Comp. 34 (1980), 333372
MSC:
Primary 65L10
MathSciNet review:
559190
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Abstract: This paper analyzes a highaccuracy approximation to the mthorder linear ordinary differential equation . At mesh points, U is the estimate of u; and U satisfies , where is a linear combination of values of U at stencil points (adjacent mesh points) and is a linear combination of values of f at J auxiliary points, which are between the first and last stencil points. The coefficients of , are obtained ``locally'' by solving a small linear system for each group of stencil points in order to make the approximation exact on a linear space S of dimension . For separated twopoint boundary value problems, U is the solution of an nbyn linear system with full bandwidth . For S a space of polynomials, existence and uniqueness are established, and the discretization error is the first divided differences of U tend to those of u at this rate. For a general set of auxiliary points one has ; but special auxiliary points, which depend upon M and the stencil points, allow larger L, up to . Comparison of operation counts for this method and five other common schemes shows that this method is among the most efficient for given convergence rate. A brief selection from extensive experiments is presented which supports the theoretical results and the practicality of the method.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005591908
PII:
S 00255718(1980)05591908
Article copyright:
© Copyright 1980
American Mathematical Society
