M. Van Daele and J. R. Cash

Deferred correction, which looks like,

() = 0          (1)
() = ()          (2)

is a widely used technique for the solution of first order systems of nonlinear two-point boundary value problems

y'(x) = f(x,y), a x b,   g(y(a), y(b)) = 0.     (3)

In an influential paper, Skeel has proven the following result. Consider the approximate numerical solution of (3) on a mesh : a = x₁ < x₂ < ... < x_N+1 = b. Denote by y the restriction of the continuous solution y(x) to the finite grid . Let be a stable numerical method and assume that the following conditions hold for the deferred correction scheme (1) , (2) : (i) | - y| = O(h^p), (ii) |(y) - (y)| = O(h^r+p) and (iii) (w) = O(h^r) for arbitrary functions w having at least r continuous derivatives. If () = () then | - y| = O(h^r+p).

In the context of two-point BVPs, can be chosen to be a Runge-Kutta methods of orders p while = -* where * is a Runge-Kutta method of order p+r. The feature that is common to all of the deferred correction schemes that have been derived so far is that r = 2 and for schemes of this type the order of accuracy is increased by 2. A sufficient condition to achieve this increase in accuracy is basically that the Runge-Kutta formulae and * should be symmetric and that they should be written in a special way that is appropriate for boundary value problems. The main reason why it is hard to get more than two orders of accuracy improvement per iteration is the difficulty in satisfying condition (iii) for r > 2.

The present talk addresses the question of whether it is possible to choose and * so that we can achieve r > 2 while maintaining the efficiency of the deferred correction scheme. For obvious reasons, we call such deferred correction schemes with r > 2 super convergent.

MINISIMPOSIUM SESSION: 8. Software issues