We show that each isolated solution, $y(t)$, of the general nonlinear two-point boundary value problem $( * ):y' = f(t,y),a < t < b,g(y(a),y(b)) = 0$ can be approximated by the (box) difference scheme $( * * ):{{[u_j - u_{j - 1} ]} / {h_j }} = f(t_{{{j - 1} 2}},{{[u_j + u_{j - 1} ]} / 2}),\, 1 \leqq j \leqq J,\, g(u_0,u_J ) = 0$. For $h = \max _{1 \leqq j \leqq J} h_j $ sufficiently small, the difference equations (**) are shown to have a unique solution $\{ u_j \} _0^J $ in some sphere about $\{ y(t_j )\} _0^J $, and it can be computed by Newton’s method which converges quadratically. If $y(t)$ is sufficiently smooth, then the error has an asymptotic expansion of the form $u_j - y(t_j ) = \sum _{v = 1}^m {h^{2v} e_v (t_j ) + O(h^{2m + 2} )} $, so that Richardson extrapolation is justified.
The coefficient matrices of the linear systems to be solved in applying Newton’s method are of order $n(J + 1)$ when $y(t) \in \mathbb{R}^n $. For separated endpoint boundary conditions: $g_1 (y(a)) = 0,\, g_2 (y(b)) = 0$ with $\dim g_1 = p,\dim g_2 = q$ and $p + q = n$, the coefficient matrices have the special block tridiagonal form $A \equiv [B_j,A_j,C_j ]$ in which the $n \times n$ matrices $B_j (C_j )$ have their last q (first p) rows null. Block elimination and band elimination without destroying the zero pattern are shown to be valid. The numerical scheme is very efficient, as a worked out example illustrates.

Get full access to this article

View all available purchase options and get full access to this article.


William B. Gragg, On extrapolation algorithms for ordinary initial value problems, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 384–403
J. W. Jerome, R. S. Varga, Generalizations of spline functions and applications to nonlinear boundary value and eigenvalue problems, Theory and Applications of Spline Functions (Proceedings of Seminar, Math. Research Center, Univ. of Wisconsin, Madison, Wis., 1968), Academic Press, New York, 1969, 103–155
Herbert B. Keller, Accurate difference methods for linear ordinary differential systems subject to linear constraints, SIAM J. Numer. Anal., 6 (1969), 8–30
Herbert B. Keller, Numerical methods for two-point boundary-value problems, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968viii+184
Herbert B. Keller, Newton's method under mild differentiability conditions, J. Comput. System Sci., 4 (1970), 15–28
Heinz-Otto Kreiss, Difference approximations for boundary and eigenvalue problems for ordinary differential equations, Math. Comp., 26 (1972), 605–624
F. Nieuwstadt, H. B. Keller, Viscous flow past circular cylinders, Internat. J. Comput. & Fluids, 1 (1973), 59–71
J. M. Ortega, W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970xx+572
J. M. Varah, On the solution of block-tridiagonal systems arising from certain finite-difference equations, Math. Comp., 26 (1972), 859–868
A. B. White, Ph.D. thesis, California Institute of Technology, Pasadena, Calif., in preparation
Richard Weiss, The application of implicit Runge-Kutta and collection methods to boundary-value problems, Math. Comp., 28 (1974), 449–464

Information & Authors


Published In

cover image SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Pages: 305 - 320
ISSN (online): 1095-7170


Submitted: 9 November 1972
Published online: 14 July 2006



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media