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Type of Document	Dissertation
Author	Scheid, Robert Elmer
URN	etd-05042006-103859
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-05042006-103859
Title	The accurate numerical solution of highly oscillatory ordinary differential equations
Degree	PhD
Option	Applied and Computational Mathematics
Advisory Committee	Advisor Name Title Heinz Kreiss Committee Chair
Keywords	none
Date of Defense	1982-03-10
Availability	restricted
Abstract NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. We consider systems of ordinary differential equations with rapidly oscillating solutions. Conventional numerical methods require an excessively small time step ([...] = [...]), where h is the step size necessary for the resolution of a smooth function of t and [...] measures the size of the large eigenvalues of the system's Jacobian). For the linear problem with well-separated large eigenvalues we introduce smooth transformations which lead to the separation of the time scales and computation with a large time step ([...] = [...]). For more general problems, including systems with weak polynomial nonlinearities, we develop an asymptotic theory which leads to expansions whose terms are suitable for numerical approximation. Resonances can be detected and resolved often with a large time step ([...] = [...]). Passage through resonance in nonautonomous systems can be resolved by a moderate time step ([...] = [...]).
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