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

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]**Carl de Boor and Blâir Swartz,*Collocation at Gaussian points*, SIAM J. Numer. Anal.**10**(1973), 582–606. MR**0373328****[3]**William B. Gragg,*On extrapolation algorithms for ordinary initial value problems*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 384–403. MR**0202318****[4]**Herbert B. Keller,*Accurate difference methods for linear ordinary differential systems subject to linear constraints*, SIAM J. Numer. Anal.**6**(1969), 8–30. MR**0253562****[5]**R. D. Russell, "Collocation for systems of boundary value problems,"*SIAM J. Numer. Anal.*(Submitted.)**[6]**R. D. Russell and L. F. Shampine,*A collocation method for boundary value problems*, Numer. Math.**19**(1972), 1–28. MR**0305607****[7]**R. D. Russell & J. M. Varah, "Equivalences in global methods for two-point boundary value problems," (In preparation.)**[8]**Martin H. Schultz,*Spline analysis*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. Prentice-Hall Series in Automatic Computation. MR**0362832****[9]**J. M. Varah,*On the solution of block-tridiagonal systems arising from certain finite-difference equations*, Math. Comp.**26**(1972), 859–868. MR**0323087**, 10.1090/S0025-5718-1972-0323087-4

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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