Date of Degree

9-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor(s)

Dennis Sullivan

Committee Members

Scott Wilson

Martin Bendersky

Subject Categories

Physical Sciences and Mathematics

Keywords

Manauri

Abstract

Lax and Richtmyer developed a theory of algorithms for linear initial value problems that guarantees, under certain circumstances, the convergence to numerical solution of initial value problem. The assumptions are first that the difference equations (algorithms) approx- imate the differential equations under study (this is called consistency) and, secondly, that the initial value problem be well-posed (which means that the solutions exist, are unique and depend continuously on initial data). Under these assumptions the stability condition (which requires that errors in the algorithm do not accumulate nor increase as one iter- ates the algorithm) is necessary and sufficient for convergence in a certain uniform sense for arbitrary initial data. In this work we will extend certain aspects of their work to the nonlinear context. We drop the PDE and the well-posedness assumptions at first and add the ”β − axioms” that will guarantee convergence [ Theorems 2 and 3 ] of algorithm orbits in a projective limit of finite dimensional spaces. A conjecture for a partial converse that some stability is a consequence of convergence for a natural class of nonlinear algorithms where the deviation of these non-linear algorithms from being linear is itself a bilinear map. When the algorithms satisfy consistency with a PDE initial value problem we obtain the definition of a new kind of numerical solution and their existence [Theorem 6] given said algorithms.

 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.