My MSc Thesis in Applied Mathematics, entitled Backward Error Analysis as a Model of Computation for Numerical Methods, was defended in Summer, 2011, under the supervision of Robert M. Corless (Professor of applied mathematics at the University of Western Ontario).
You can download my MSc Thesis here.
The referees were Robert W. Batterman (Professor of philosophy at the University of Pittsburgh), David Jeffrey (Professor of applied mathematics at the University of Western Ontario) and Stephen M. Watt (Professor of computer science at the University of Western Ontario).
This thesis delineates a generally applicable perspective on numerical methods for scientific computation called residual-based a posteriori backward error analysis, based on the concepts of condition, backward error, and residual, pioneered by Turing and Wilkinson. The basic underpinning of this perspective, that a numerical method's errors should be analyzable in the same terms as physical and modelling errors, is readily understandable across scientific fields, and it thereby provides a view of mathematical tractability readily interpretable in the broader context of mathematical modelling. It is applied in this thesis mainly to numerical solution of differential equations. We examine the condition of initial-value problems for ODEs and present a residual-based error control strategy for methods such as Euler's method, Taylor series methods, and Runge-Kutta methods. We also briefly discuss solutions of continuous chaotic problems and stiff problems.
backward error analysis, condition number, residual, numerical stability, floating-point arithmetic, numerical solution of ordinary differential equations, stiffness, local error, Taylor series method, continuous Runge-Kutta method