A high-order difference method for differential equations

Lynch, Robert E.; Rice, John R.

doi:10.1090/S0025-5718-1980-0559190-8

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 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.

References

Garrett Birkhoff and Carl R. De Boor, Piecewise polynomial interpolation and approximation, Approximation of Functions (Proc. Sympos. General Motors Res. Lab., 1964) Elsevier Publ. Co., Amsterdam, 1965, pp. 164–190. MR 0189219
Garrett Birkhoff and Gian-Carlo Rota, Ordinary differential equations, 2nd ed., Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1969. MR 0236441

The Effect on Accuracy of the Placement of Auxiliary Points in the HODIE Method for the Helmholtz Problem

Lothar Collatz, The numerical treatment of differential equations. 3d ed, Die Grundlehren der mathematischen Wissenschaften, Band 60, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. Translated from a supplemented version of the 2d German edition by P. G. Williams. MR 0109436
H. B. Curry and I. J. Schoenberg, On Pólya frequency functions. IV. The fundamental spline functions and their limits, J. Analyse Math. 17 (1966), 71–107. MR 218800, DOI 10.1007/BF02788653
Carl de Boor and Robert E. Lynch, On splines and their minimum properties, J. Math. Mech. 15 (1966), 953–969. MR 0203306
Carl de Boor and Blâir Swartz, Collocation at Gaussian points, SIAM J. Numer. Anal. 10 (1973), 582–606. MR 373328, DOI 10.1137/0710052

The Construction of Finite Difference Approximations to Ordinary Differential Equations

Eusebius J. Doedel, The construction of finite difference approximations to ordinary differential equations, SIAM J. Numer. Anal. 15 (1978), no. 3, 450–465. MR 483481, DOI 10.1137/0715029

Tchebycheff Systems

With Applications in Analysis and Statistics

The HODIE Method

A Brief Introduction with Summary of Computational Properties

Robert E. Lynch and John R. Rice, High accuracy finite difference approximation to solutions of elliptic partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 6, 2541–2544. MR 496774, DOI 10.1073/pnas.75.6.2541

Accurate Finite Difference Approximation to Solutions of the Poisson Equation in Three Variables

Discretization Error Finite Difference Approximation to Solutions of the Poisson Equation in Three Variables

Robert E. Lynch and John R. Rice, High accuracy finite difference approximation to solutions of elliptic partial differential equations, Proc. Nat. Acad. Sci. U.S.A. 75 (1978), no. 6, 2541–2544. MR 496774, DOI 10.1073/pnas.75.6.2541
Robert E. Lynch and John R. Rice, The Hodie method and its performance for solving elliptic partial differential equations, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 143–175. MR 519061
M. R. Osborne, Minimising truncation error in finite difference approximations to ordinary differential equations, Math. Comp. 21 (1967), 133–145. MR 223107, DOI 10.1090/S0025-5718-1967-0223107-X
James L. Phillips and Richard J. Hanson, Gauss quadrature rules with $B$-spline weight functions, Math. Comp. 28 (1974), no. 126, loose microfiche suppl, A1–C4. MR 343551, DOI 10.2307/2005948
Carl de Boor (ed.), Mathematical aspects of finite elements in partial differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR 0349031
R. D. Russell and L. F. Shampine, A collocation method for boundary value problems, Numer. Math. 19 (1972), 1–28. MR 305607, DOI 10.1007/BF01395926
R. D. Russell and J. M. Varah, A comparison of global methods for linear two-point boundary value problems, Math. Comput. 29 (1975), no. 132, 1007–1019. MR 0388788, DOI 10.1090/S0025-5718-1975-0388788-3
Robert D. Russell, A comparison of collocation and finite differences for two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 1, 19–39. MR 451745, DOI 10.1137/0714003
Carl de Boor (ed.), Mathematical aspects of finite elements in partial differential equations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Publication No. 33 of the Mathematics Research Center, The University of Wisconsin-Madison. MR 0349031