Parallel methods for the numerical solution of ordinary differential equations

Abstract

We study time parallelism for the numerical solution of nonstiff ordinary differential equations. Stability and accuracy are the two main considerations in deriving good numerical o.d.e. methods. However, existing parallel methods have poor stability properties in that their stability regions are smaller than those of good sequential methods of the same order. In this thesis we present a precise understanding of how stability limits the potential of parallelism in o.d.e.'s. We propose a fairly specific approach to construct good parallel methods--we consider zero-stable parallel methods whose stability polynomials are perfect powers of those of simple methods with good stability regions. Based on this approach we derive new efficient parallel methods. The proposed families of block methods have stability regions which do not change as the order increases. These new methods have much better stability properties than the Adams PECE methods of the same order. The above perfect power stability polynomial approach can also be extended to multi-block methods.