Economical error estimates for block implicit methods for ODEs via deferred correction

Deferred correction is a widely used tool for improving the numerical approximation to the solution of ODE problems [J.R. Cash, WSSIA 2 (1993) 113; J.R. Cash, M.H. Wright, SIAM J. Sci. Statist. Comput. 12 (1991) 971; M. Lentini, V. Pereyra, Math. Comp. 28 (1974) 981; B. Lindberg, BIT 20 (1980) 486; V. Pereyra, Numer. Math. 8 (1966) 376; V. Pereyra, Numer. Math. 10 (1967) 316; H.J. Stetter, Numer. Math. 29 (1978) 425; H.J. Stetter, in: Lecture Notes in Math. vol. 630, Springer, 1978, pp. 245–258; R.D. Skeel, SIAM J. Numer. Anal. 19 (1981) 171; R.D. Skeel, Numer. Math. 48 (1986) 1; P. Zadunaisky, Numer. Math. 27 (1976) 21]. Indeed, it allows to estimate the error due to the use of discrete methods. Such an estimate may be a global one, in the case of continuous BVPs, or a local one, when IVPs are to be approximated [L. Brugnano, in: Lecture Notes in Math., vol. 1196, Springer, 1997, pp. 78–89; L. Brugnano, D. Trigiante, Solving Differential Problems by Multistep Initial and Boundary Value Methods, Gordon and Breach, 1998]. Recently, it has been implemented in the computational code BiM [L. Brugnano, C. Magherini, J. Comput. Appl. Math. 164–165 (2004) 145, web page: http://math.unifi.it/~brugnano/BiM/index.html] for the numerical solution of stiff ODE-IVPs. In this paper we analyze deferred correction in connection with the methods used in that code, resulting in an overall simplification of the procedure, due to the properties of the underlying methods. The analysis is then extended to more general methods.