Spatial error estimation for collocation solutions of differential equations

Abstract

Computational Science is now a central component of all scientific investigation, along with the traditional modes of experimental and theoretical investigation. Computational Science involves the development and solution of mathematical models, i.e., systems of equations, that represent approximations to real world phenomenon in a wide variety of scientific areas. These mathematical models typically do not have closed form solutions and thus the models are solved using computational software to obtain approximate solutions. Since these solutions are approximate, the question that must be addressed is “How Good is the Computed Solution?”. This question is answered for a given numerical solution through the computation of a good quality error estimate. This thesis will discuss current work on answering this question in the area of computational methods for differential equations that depend on time and/or one or more spatial dimensions. We will describe the use of collocation, a general numerical method that can be used to obtain approximate solutions for a wide range of problem classes, as well as our recent work in the development of efficiently computable error estimates for collocation solutions based on special types of interpolants. We provide results from numerical experiments to demonstrate the effectiveness of our approach.