Studies in Numerical Solution of Ordinary Differential Equations

本项目的目的是研究用于连续和离散波形松弛迭代的各种加速技术。这些技术包括指数、多项式或Toeplitz预条件和/或系统的重叠组件。我们还将研究常微分方程组的新型数值方法的构造和实现，这些方法是一般线性方法的特例。这种方法适用于顺序或并行计算环境中的非刚性或刚性微分系统。与经典方法相比，这些方法也有更高的潜力来保持解的定性性质。这一潜力将被用来集成波传播问题和吸收边界条件。在这一建议的支持下的研究将导致现代软件的发展，以解决许多由大型微分系统控制的科学和工程问题。该软件将基于主要研究人员和他的同事在过去几年中开发的新的数值方法，并将利用现代计算机体系结构，将潜在的大问题分成较小的子系统或子问题，然后通过将它们分配到不同的处理器来并行解决。对许多问题的数值实验，例如声波传播、化学反应、汽车通过的板的运动、神经脉冲机制的突变模型、描述可形成簇的大量粒子的系统动力学的模型以及反应扩散系统等，表明这些新方法比目前用于这些问题的方法更有效、更健壮，更适合于利用最新的计算机体系结构。

JackiewiczIt is the purpose of this project to study various acceleration techniques for continuous and discrete waveform relaxationiterations. These techniques include exponential, polynomial orToeplitz preconditioning and/or overlapping components of the system. We will also study the construction and implementation of novel numerical methods for ordinary differential equationswhich are special cases of general linear methods. Such methods are appropriate for nonstiff or stiff differential systems in asequential or parallel computing environment. These methods havealso higher potential to preserve qualitative properties of the solution than classical methods. This potential will be exploitedto integrate wave propagation problems with absorbing boundaryconditions.The research supported by this proposal will lead to the developmentof modern software for the solution of many problems in science andengineering which are governed by large differential systems. Thissoftware will be based on novel numerical methods developed in thepast few years by the principal investigator and his coworkers and will utilize modern computer architecture by splitting the underlyinglarge problem into smaller subsystems or subproblems which can then be solved in parallel by allocating them to different processors. Numerical experiments on many problems, for example, acoustic wavepropagation, chemical reactions, movements of a plate under the loadof a car passing through it, catastrophe model for the nerve impulsemechanism, model that describes the dynamics of a system with largenumber of particles which can form clusters, and reaction-diffusionsystems, indicate that these novel methods are more efficient and robust and better suited to take advantage of the state-of-the-artcomputer architecture than the methods which are currently in use forthese problems.