Document Type
Dissertation
Date of Degree
Spring 2011
Degree Name
PhD (Doctor of Philosophy)
Degree In
Applied Mathematical and Computational Sciences
First Advisor
Laurent O. Jay
First Committee Member
Rodica Curtu
Second Committee Member
Luca Dieci
Third Committee Member
Weimin Han
Fourth Committee Member
David Stewart
Abstract
Differential-algebraic equations (DAEs) consist of mixed systems of ordinary differential equations (ODEs) coupled with linear or nonlinear equations. Such systems may be viewed as ODEs with integral curves lying in a manifold. DAEs appear frequently in applications such as classical mechanics and electrical circuits. This thesis concentrates on systems of index 2, originally index 3, and mixed index 2 and 3.
Fast and efficient numerical solvers for DAEs are highly desirable for finding solutions. We focus primarily on the class of Gauss-Lobatto SPARK methods. However, we also introduce an extension to methods proposed by Murua for solving index 2 systems to systems of mixed index 2 and 3. An analysis of these methods is also presented in this thesis. We examine the existence and uniqueness of the proposed numerical solutions, the influence of perturbations, and the local error and global convergence of the methods.
When applied to index 2 DAEs, SPARK methods are shown to be equivalent to a class of collocation type methods. When applied to originally index 3 and mixed index 2 and 3 DAEs, they are equivalent to a class of discontinuous collocation methods. Using these equivalences, (s,s)--Gauss-Lobatto SPARK methods can be shown to be superconvergent of order 2s.
Symplectic SPARK methods applied to Hamiltonian systems with holonomic constraints preserve well the total energy of the system. This follows from a backward error analysis approach. SPARK methods and our proposed EMPRK methods are shown to be Lagrange-d'Alembert integrators.
This thesis also presents some numerical results for Gauss-Lobatto SPARK and EMPRK methods. A few problems from mechanics are considered.
Keywords
Differential-Algebraic Equations, Gauss-Lobatto Coefficients, Holonomic Constraints, Lagrangian Systems, Nonholonomic Constraints, Runge-Kutta Methods
Pages
viii, 244 pages
Bibliography
Includes bibliographical references (pages 243-244).
Copyright
Copyright 2011 Scott Joseph Small