A comparison of some numerical methods for two-point boundary value problems

Author:
James M. Varah

Journal:
Math. Comp. **28** (1974), 743-755

MSC:
Primary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1974-0373300-4

MathSciNet review:
0373300

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Abstract: In this paper we discuss and compare two useful variable mesh schemes for linear second-order two-point boundary value problems: the midpoint rule and collocation with cubic Hermite functions. We analyze the stability of the block-tridiagonal factorization for solving the linear systems, compare the amount of computer time required, and test the methods on some particular numerical problems.

**[1]**G. F. Carrier, "Singular perturbation theory and geophysics,"*SIAM Rev.*, v. 12, 1970, pp. 175-193.**[2]**G. de Boor & B. Swartz, "Collocation at Gaussian points,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 582-606. MR**0373328 (51:9528)****[3]**W. B. Gragg, "On extrapolation algorithms for ordinary initial value problems,"*J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.*, v. 2, 1965, pp. 384-403. MR**34**#2191. MR**0202318 (34:2191)****[4]**H. B. Keller, "Accurate difference methods for linear ordinary differential systems subject to linear constraints,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 8-30. MR**40**#6776. MR**0253562 (40:6776)****[5]**R. D. Russell, "Collocation for systems of boundary value problems,"*SIAM J. Numer. Anal.*(Submitted.)**[6]**R. D. Russell & L. F. Shampine, "A collocation method for boundary value problems,"*Numer. Math.*, v. 19, 1972, pp. 1-28. MR**46**#4737. MR**0305607 (46:4737)****[7]**R. D. Russell & J. M. Varah, "Equivalences in global methods for two-point boundary value problems," (In preparation.)**[8]**M. H. Schultz,*Spline Analysis*, Prentice-Hall, Englewood Cliffs, N.J., 1973. MR**0362832 (50:15270)****[9]**J. M. Varah, "On the solution of block-tridiagonal systems arising from certain finite-difference equations,"*Math. Comp.*, v. 26, 1972, pp. 859-868. MR**0323087 (48:1445)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0373300-4

Keywords:
Two-point boundary value problems,
collocation,
Galerkin method,
finite-differences,
operation counts,
numerical stability

Article copyright:
© Copyright 1974
American Mathematical Society