Parallel Numerical Methods For Initial Value Problems

该项目将探索新发展的波形共轭方向法并行求解初值问题(IVP)的理论和实践方面。波形松弛(WR)是一种基于波形的迭代技术，是一种日益流行的IVP并行求解方法。共轭方向波形法极大地改善了用于MOS器件暂态模拟的WR的收敛特性。该项目将通过制定完善的趋同估计以及建立和测试并行执行，为更广泛地应用这些方法奠定基础。应用于线性和非线性常微分方程组和微分方程组的共轭方向波形法将寻求新的收敛结果。还将研究离散化和多速率积分的效果。点式和基于波形的IVP解算器将在ALEWAY模拟器ALIM和ALLEWAY并行计算机上实现。我们将比较它们在不同硬件配置上针对各种ODE和DAE问题的行为。波形共轭方向法似乎是一类高效实用的并行技术，具有广泛的应用前景。理论结果和实际应用将证明它们的价值和适用性。

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.