Dynamics of numerics of linearized collocation methods

Khumalo, Melusi (1997) Dynamics of numerics of linearized collocation methods. Doctoral (PhD) thesis, Memorial University of Newfoundland.

[English] PDF (Migrated (PDF/A Conversion) from original format: (application/pdf)) - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. Download (13MB) [English] PDF - Accepted Version Available under License - The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. (Original Version)

Abstract

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.

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