# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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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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