Projective Methods for Sti Di erential Equations: problems with gaps in their eigenvalue spectrum

We show that there exist classes of explicit numerical integration methods that can handle very sti problems if the eigenvalues are separated into two clusters, one containing the \sti ", or fast components, and one containing the slow components. These methods have large average step sizes relative to the fast components. Conventional implicit methods involve the solution of non-linear equations at each step, which for large problems requires signi cant communication between processors on a multiprocessor machine. For such problems the methods proposed here have signi cant potential for speed improvement.