A digraph theoretic parallelism in block methods

Abstract

In the exploitation of sequential computation expertise in the parallel solution of initial value problems in ordinary differential equations three notable directions have evolved, these are listed to include, parallelism across the method, parallelism across the steps and parallelism across the system. In this paper, parallel block methods which are methods of the first approach above are considered to take advantage of parallelism across the machine by employing the theoretic concepts of digraphs. This has been used by Jackson, Norsett and Iserles in their search for parallelism in Runge–Kutta Methods (RKM). This approach presents a higher prospective advantage for block methods arising from the fact that for a given block dimension, a block method can attain higher order and may have a wider stability region over a RKM with a comparable number of stages. Furthermore, the notion of isomorphic methods are introduced, an example of which is the embedded RKM pairs, more are presented in our generalized formulation of multi-block methods.