Numerical Methods for Sixth-Order Boundary Value Problems

Abstract

A second-order convergent finite difference method is discussed for the numerical solution of the special nonlinear sixth-order boundary-value problem w^(vi) (x) — f(x, w), a < x < b, w(a) = A₀, w″(a) = A₂, w^(iv)(a) — A₄, w(b) — B₀, w″(b) — B₂, w^(iv)(b) — B₄. Adaptation to the special linear problem with differential equation w^(vi) (x) = q(x)w(x) + r(x) is considered briefly. Fourth- and sixth-order convergence is obtained by using the numerical method on two and three grids, respectively, and taking a linear combination of the individual results relating to the extra grid(s). Special formulas are developed for application to grid points adjacent to the boundaries x — a and x — b, the first two terms of the local truncation errors of these formulas being the same as those of the second-order method used at other points of each grid.

Adaptation to general sixth-order boundary-value problems is illustrated by reference to the Bénard layer problem, which has differential equation -w^(vi)(x) + 3A²w^(iv)(x) — 3A⁴w″(x) + A⁶w(x) — RA² (1-x²)w(x) + f(x,w) = 0, and which arises in astrophysics.