VPW: Waveform Relaxation Method

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The advent of new generation of massively parallel computers has caused previously efficient numerical algorithms to be reexamined. Massively parallel computers are comprised of many hundreds, sometimes thousands of processors. For a numerical algorithm to fully exploit the power of such machines it must be able to be 'decomposed' into largely independent pieces which can be distributed across the available computer processors. The waveform relaxation method is such a method for solving systems of ordinary differential equations and time-dependent partial differential equations. In addition to its advantages in parallel processing, it also allows different integration step sizes to be used for different subsystems, resulting in substantial savings in computation for some applications. The theoretical and experimental studies suggest that the waveform relaxation method has a great potential for developing reliable, scalable efficient parallel algorithms for solving a wide range of differential equations. The objective of this research is to further investigate the nature of the waveform relaxation method, and develop fast scalable parallel algorithms for solving certain type of ordinary differential equations and time-dependent partial differential equations. The long term goal of this research is to make the developed algorithms applicable to real-world computational problems, such as the real time integration of torpedo or missile trajectories, modeling of earthquake effects on buildings, and simulation of fluid-structure interaction. This work will involve the algorithm design, theoretical analysis with a possible focus on the search of optimal convergence rate, and numerical testing on sequential and parallel computers. Interactive activities include: teaching a course at the graduate level on numerical linear algebra, numerical partial differential equation, or on parallel numerical computation; and leading an interdisciplinary research seminar seri es on scientific computation and applications to which both local and national researchers will be invited to participate.