The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations

Citation

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. https://resolver.caltech.edu/CaltechETD:etd-05042006-103859

Abstract

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)).

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