A two - grid, fourth order method for nonlinear fourth order boundary value problems

Abstract

A fourth order convergent finite difference method is developed for the numerical solution of the nonlinear fourth order boundary value problem y(iv)(x) = f(x,y), a<x <b, y(a) = A0 , y"(a) = B0 , y(b) = A1, y" (b) = B1 . The method is based on a second order convergent method which is used on two grids, fourth order convergence being obtained by considering a linear combination of the individual results relating to the two grids. Special formulas are developed for application to grid points adjacent to the boundaries x = a and x = b , the principal parts of the local truncation errors of these formulas being the same as that of the second order method used at other points of each grid. Modifications to these special formulas are noted for problems with boundary conditions of the form y (a) = Ao , y'(a) = Co , y(b) = A1, y'(b) =c1.