We show that there exist classes of explicit numerical integration methods that can handle very stiff problems if the eigenvalues are separated into two clusters, one containing the "stiff" 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 nonlinear equations at each step, which for large problems requires significant communication between processors on a multiprocessor machine. For such problems the methods proposed here have significant potential for speed improvement.