Blended Implementation of Block Implicit Methods for ODEs

In this paper we further develop a new approach for naturally defining the nonlinear splittings needed for the implementation of block implicit methods for ODEs, which has been considered by Brugnano [J. Comput. Appl. Math. 116 (2000) 41] and by Brugnano and Trigiante [in: Recent Trends in Numerical Analysis, Nova Science, New York, 2000, pp. 81-105]. The basic idea is that of defining the numerical method as the combination (blending) of two suitable component methods. By carefully choosing such methods, it is shown that very efficient implementations can be obtained. Moreover, some of them, characterized by a diagonal splitting, are well suited for parallel computers. Some numerical tests comparing the performances of the proposed implementation with other existing ones are also presented, in order to make evident the potential of the approach.