Parallel Adams methods

In the literature, various types of parallel methods for integrating nonstiff initial-value problems for first-order ordinary differential equation have been proposed. The greater part of them are based on an implicit multistage method in which the implicit relations are solved by the predictor-corrector (or fixed point iteration) method. In the predictor-corrector approach the computation of the components of the stage vector iterate can be distributed over s processors, where s is the number of implicit stages of the corrector method. However, the fact that after each iteration the processors have to exchange their just computed results is often mentioned as a drawback, because it implies frequent communication between the processors. Particularly on distributed memory computers, such a fine grain parallelism is not attractive. An alternative approach is based on implicit multistage methods which are such that the implicit stages are already parallel, so that they can be solved independently of each other. This means that only after completion of a step, the processors need to exchange their results. The purpose of this paper is the design of a class of parallel methods for solving nonstiff IVPs. We shall construct explicit methods of order k+1 with k parallel stages where each stage equation is of Adams-Bashforth type and implicit methods of order k+2 with k parallel stages which are of Adams- Moulton type. The abscissae in both families of methods are proved to be the Lobatto points, so that the Adams- Bashforth type method can be used as a predictor for the Adams-Moulton type corrector.