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 $( * * ):{{[u_j - u_{j - 1} ]} / {h_j }} = f(t_{{{j - 1} 2}} ,{{[u_j + u_{j - 1} ]} / 2}),\, 1 \leqq j \leqq J,\, g(u_0 ,u_J ) = 0$. 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.