Applied Numerical Mathematics 11 (1993) 225-239 North-Holland 225 APNUM 367 Boundary value methods and BV-stability in the solution of initial value problems * L. Lopez and D. Trigiante Dipartimento di Matematica, Uniuersiti di Bari, Via G. Fortunato 70125, Bari, Italy Abstract Lopez, L. and D. Trigiante, Boundary value methods and BV-stability in the solution of initial value problems, Applied Numerical Mathematics 11 (1993) 225-239. In this paper we consider boundary value techniques based on a three-term numerical method for solving initial value problems. The notions of BV-stability and BV-relative stability are introduced in order to clarify the conditions that a three-term scheme must satisfy for solving efficiently initial value problems. In particular we investigate the BV-stability of boundary value methods based on the mid-point rule, on the Simpson method, and on an Adams-type method. The problem of approximating the solution at the final point is approached and an error estimate at this point is given. Among the main features of the boundary value methods studied there is the possibility of employing the same method for an initial value problem with increasing and decreasing modes and the possibility of implementing efficiently boundary value methods on parallel computers. zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR 1. Initial and boundary value problems The idea of transforming an initial value problem into a boundary value problem (BVP) goes back to J.P. Miller (1952) and F.W. Olver (1967). It is based on the fact that if an equation (continuous or discrete) has a dominant and subdominant solution, in general the solution of the boundary value problem tends to approximate the subdominant one with the same initial point. The proof of this fact was given by Olver in the discrete case and can be found in many places (see [5,13,14]). The idea was stated first for discrete problems and used to compute solutions of unstable recurrence relations (Bessel functions). A good survey paper on this question is Gautschi’s [8]. Carasso in [4] applied the idea to solving parabolic equations with a steady-state solution, while its application to numerical methods for ODES was considered extensively by Cash in [5]. More recently Axelsson and Verwer treated the same problem by applying a boundary value method based on the mid-point rule (see [l]). Correspondence to: D. Trigiante, Dipartimento di Matematica, Universith di Bari, Via G. Fortunato 70125, Bari, Italy. * Work supported by “Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo” of C.N.R. 0168-9274/93/$05.00 0 1993 - El sevier Science Publishers B.V. All rights reserved