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The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations


Scheid, Robert Elmer, Jr (1982) The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4JVY-JB67.


We consider systems of ordinary differential equations with rapidly oscillating solutions. Conventional numerical methods require an excessively small time step (Δt = 0(εh), where h is the step size necessary for the resolution of a smooth function of t and 1/ε 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 (Δt = 0(h)). 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 (Δt = 0(h)). Passage through resonance in nonautonomous systems can be resolved by a moderate time step (Δt = 0(√εh)).

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Oscillatory Differential Equations
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kreiss, Heinz-Otto
Thesis Committee:
  • Kreiss, Heinz-Otto (chair)
  • Cohen, Donald S.
  • Keller, Herbert Bishop
  • Tadmor, E.
  • Caughey, Thomas Kirk
Defense Date:10 March 1982
Funding AgencyGrant Number
Office of Naval Research (ONR)N00014-80-C-0076
Record Number:CaltechETD:etd-05042006-103859
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1601
Deposited By: Imported from ETD-db
Deposited On:05 May 2006
Last Modified:26 Jun 2020 00:42

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