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.