Parallel Numerical Methods For Initial Value Problems

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This project will explore theoretical and practical aspects of newly-developed waveform conjugate-direction methods for parallel solution of initial value problems (IVPs). Waveform relaxation (WR), a waveform-based iterative technique, is an increasingly popular parallel solution approach for IVPs. Conjugate-direction waveform methods have dramatically improved the convergence properties of WR for transient simulation of MOS devices. The project will establish the basis for broader application of these methods by developing refined convergence estimates and by building and testing a parallel implementation. New convergence results will be sought for conjugate-direction waveform methods applied to linear and nonlinear systems of ODEs and DAEs. The effects of discretization and multirate integration will also be studied. Pointwise and waveform-based IVP solvers will be implemented on ASIM, the Alewife simulator, and on the Alewife parallel computer. Their behavior for a variety of ODE and DAE problems on various hardware configuration will be compared. Waveform conjugate-direction methods appear to be an efficient and practical class of parallel techniques for a variety of applications. Together, the theoretical results and practical implementations will demonstrate their value and applicability.