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Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems

Keller, Herbert B. (1974) Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems. SIAM Journal on Numerical Analysis, 11 (2). pp. 305-320. ISSN 0036-1429. doi:10.1137/0711028.

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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-½),[u_j + u_(j-1)]/2), 1 ≦ j ≦ J, g(U_0,U_J) = O. For h = max_(1 ≦j≦J)h_j sufficiently small, the difference equations (**) are shown to have a unique solution {U_j}^J_0} in some sphere about {y(t_j)}^J_0, 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) = Σ^(m)_(v=1) 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 + l) when y(t) ∈ 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 ≡ [B_j, A_j, C_j] in which the n x 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.

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Additional Information:© 1974 Society for Industrial and Applied Mathematics. Received by the editors November 9, 1972, and in revised form March 1, 1973. This work was supported by the Atomic Energy Commission under Contract AT(04-3)-767, Project Agreement no. 12.
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Atomic Energy CommissionAT(04-3)-767
Issue or Number:2
Record Number:CaltechAUTHORS:20120808-142816266
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Official Citation:Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems Keller, H. SIAM Journal on Numerical Analysis 1974 11:2, 305-320
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:33028
Deposited On:08 Aug 2012 22:59
Last Modified:09 Nov 2021 21:31

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