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

DOI:
https://doi.org/10.1090/S0025-5718-1980-0559190-8

MathSciNet review:
559190

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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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Article copyright:
© Copyright 1980
American Mathematical Society