# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## A high-order difference method for differential equationsHTML articles powered by AMS MathViewer

by Robert E. Lynch and John R. Rice
Math. Comp. 34 (1980), 333-372 Request permission

## 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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