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Mathematics of Computation

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A high-order difference method for differential equations

Authors: Robert E. Lynch and John R. Rice
Journal: Math. Comp. 34 (1980), 333-372
MSC: Primary 65L10
MathSciNet review: 559190
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Abstract: This paper analyzes a high-accuracy approximation to the mth-order linear ordinary differential equation $Mu = f$. At mesh points, U is the estimate of u; and U satisfies ${M_n}U = {I_n}f$, where ${M_n}U$ is a linear combination of values of U at $m + 1$ stencil points (adjacent mesh points) and ${I_n}f$ is a linear combination of values of f at J auxiliary points, which are between the first and last stencil points. The coefficients of ${M_n}$, ${I_n}$ 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 $L + 1$. For separated two-point boundary value problems, U is the solution of an n-by-n linear system with full bandwidth $m + 1$. For S a space of polynomials, existence and uniqueness are established, and the discretization error is $O({h^{L + 1 - m}})$ the first $m - 1$ divided differences of U tend to those of u at this rate. For a general set of auxiliary points one has $L = J + m$; but special auxiliary points, which depend upon M and the stencil points, allow larger L, up to $L = 2J + m$. 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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Article copyright: © Copyright 1980 American Mathematical Society