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 - uj - 1 ] / hⱼ = f(t_j - 1 2 ,[uⱼ + uj - 1 ] / 2),\, 1 j J,\, g(u₀ ,uJ ) = 0. For h = max 1 j J 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 ) = ∑ v = 1ᵐ h²ᵛ eᵥ (tⱼ ) + O(h2m + 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.