Accurate Difference Methods for Nonlinear Two-Point Boundary Value Problems

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 (* *): {uⱼ - u₉ - ₁ / hⱼ } = f (t{₉ - ₁ 2}, {uⱼ + u₉ - ₁ / 2}), \, 1 j J, \, g (u₀, uJ) = 0. For h = ₁ ₉ ₉ hⱼ sufficiently small, the difference equations (**) are shown to have a unique solution \ uⱼ \ ₀J in some sphere about \ y (tⱼ) \ ₀J, 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 uⱼ - y (tⱼ) = ₕ = ₁ᵐ h^{2v eᵥ (tⱼ) + 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) Rⁿ. For separated endpoint boundary conditions: g₁ (y (a) ) = 0, \, g₂ (y (b) ) = 0 with g₁ = p, g₂ = q and p + q = n, the coefficient matrices have the special block tridiagonal form A Bⱼ, Aⱼ, Cⱼ in which the n n matrices Bⱼ (Cⱼ) 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.