Some P(EC)mE methods for the parallel solution of ODEs

Stability and efficiency are two important considerations in deriving good numerical methods for ODEs. The underlying challenge is to increase the stability region while maintaining or even improving efficiency. In this paper, we consider this problem within the context of parallel solution methods. A family of methods which we call parallel block predictor-corrector methods is proposed. Members of this family are characterized by iterated corrector evaluations carried out in two adjacent blocks. Stability properties of these methods are analyzed and compared with some existing block-based parallel predictor-corrector methods. Performance of two particular members of the family is compared with existing block-based parallel predictor-corrector methods, using solution formulas which extend over a range of integration orders and which use various numbers of processors. The results obtained from a stability analysis and from a collection of numerical experiments indicate that the proposed methods have the potential to balance stability properties and efficiency in parallel computer systems.