Studies in Numerical Solution of Ordinary Differential Equations

点击翻译按钮获取中文摘要

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.