Defect estimation in Adams PECE codes

Many modern codes for solving the nonstiff initial value problem $y'(x) - f(x,y(x)) = 0,y(a)$ given, $a leqq x leqq b$, produce, in addition to a discretised solution, a function $p(x)$ that approximates $y(x)$ over $[a,b]$. The associated defect $delta (x): = p'(x) - f(x,p(x))$ is a natural measure of the error. In this paper the problem of reliably estimating the defect in Adams PECE methods is considered. Attention is focused on the widely used Shampine-Gordon variable order, variable step code fitted with a continuously differentiable interpolant $p(x)$ due to Watts and Shampine [SIAM .J. Sci. Statist. Comput, 7 (1986), pp. 334-345]. It is shown that over each step an asymptotically correct estimate of the defect can be obtained by sampling at a single, suitably chosen point. It is also shown that a valid "free" estimate can be formed without recourse to sampling. Numerical results are given to support the theory.