Waveform methods for space and time parallelism

Waveform methods are a form of generalized Picard iteration developed by electrical engineers for the numerical solution of the large systems of ordinary differential equations that arise in circuit simulation for VLSI design. Their effectiveness is principally due to computational considerations - disk swapping, simple organization of calculations for components with very different frequency spectra and potential use of parallelism. In this paper we first examine the rate of convergence of simple extension of Picard methods suitable for parallel computation. For numerical computation, the differential equation must be replaced by a finite-dimensional equation for an approximation to its solution. The second part of the paper examines the application of the iterative methods directly to finite-difference approximations of the original differential equation. This permits a larger class of methods than obtained by discretizing the differential equations satisfied by successive iterates.