Dynamics of numerics of linearized collocation methods

• Main Author: Khumalo, Melusi, 1966-
• Other Authors: Memorial University of Newfoundland. Dept. of Mathematics and Statistics
• Format: Thesis
• Language: English
• Published: 1997
• Subjects: Collocation methods; Bifurcation theory; Newfoundland studies; University of Newfoundland
• Online Access: http://collections.mun.ca/cdm/ref/collection/theses3/id/120562

Thesis (Ph. D.), Memorial University of Newfoundland, 1998. Mathematics and Statistics Bibliography: leaves 150-155 Many ordinary differential equations that describe physical phenomena possess solutions that cannot be obtained in closed form. To obtain the solutions to these systems, the use of numerical schemes is unavoidable. Traditional numerical analysis concerns itself with obtaining error bounds within finite closed time intervals: however, the study of asymptotic or long term behaviour of solutions generated by numerical schemes has attracted a lot of interest in recent years. It is now well established that numerical schemes for nonlinear autonomous differential equations can admit asymptotic solutions which do not correspond to those of the ODE. -- This thesis studies linearized one-point collocation methods, contributing to this important investigation by considering bifurcation phenomena in autonomous ODEs and studying the dynamics of the methods for nonau- tonomous ODEs. -- Using the theory of normal forms, it is established that the common codimension-1 bifurcations that exist in continuous dynamical systems will occur in the methods at the same phase space location. However, the methods can exhibit period doubling bifurcations, which are necessarily spurious. They also introduce a singular set. which drastically affects the global dynamics of the methods. -- The technique of stroboscopic sampling of the numerical solution is used to study the dynamics of nonautonomous ODEs with periodic solutions, and conditions under which the methods have a unique periodic solution that is asymptotically stable, are stated explicitly. A link between these conditions and nonautonomous linear and nonlinear stability theory is established.