Waveform iteration and the shifted picard splitting

The theme of this paper is that the primary computational bottleneck in the solution of stiff ordinary differential equations (ODEs) and the parallel solution of nonstiff ODEs is the implicitness of the ODE rather than the approximation of the integration process (or in conventional terminology, numerical stability rather than accuracy), and therefore it may be fruitful to apply (at least conceptually) the iterative techniques needed to overcome implicitness in continuous time, before discretization—to waveforms rather than values at a point in time. Several classical iterations, based on splitting, are discussed, but the emphasis is on those not based on a partitioning of the ODE system. The shifted Picard iteration is proposed as a compromise between the cheap but slow Picard iteration and the fast but expensive Newton iteration. By varying the shift parameter from one iteration to the next, a good rate of convergence seems possible. As an alternative, the author also examines the more classical acceleration technique applied to the Picard iteration. Some experimental results are given. However, the practical aspects of discretization are beyond the scope of this paper.