Parallel block predictor-corrector methods for the numerical solution of ODE's.

Abstract

Stability and efficiency (i.e. derivative function evaluations per processor) are the two main considerations in deriving good numerical methods for ODE's. The underlying challenge is to increase the stability region while maintaining or even improving efficiency. To achieve this, some extensions of predictor-corrector based methods, which apply a fixed number of corrector iterations, are considered. This thesis studies two particular members of a family of methods called the Parallel Block Predictor-Corrector Family, which are based on these extensions. These two members are called PBPC/2 and PBPC/3. They 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 the PBPC/2 and PBPC/3 methods and these existing block-based parallel predictor-corrector methods is compared using solution formulas which extend over a range of integration orders and which use various number of processors. The results obtained from a stability analysis and from a collection of numerical experiments indicate that the proposed methods provide a potential opportunity to balance stability properties and efficiency in the parallel computer systems.