## A high-order difference method for differential equations

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

## Abstract:

This paper analyzes a high-accuracy approximation to the*m*th-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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## Additional Information

- © Copyright 1980 American Mathematical Society
- Journal: Math. Comp.
**34**(1980), 333-372 - MSC: Primary 65L10
- DOI: https://doi.org/10.1090/S0025-5718-1980-0559190-8
- MathSciNet review: 559190