The Accurate Numerical Solution of Highly Oscillatory Ordinary Differential Equations

Oscillatory Differential Equations

Scheid, Robert Elmer, Jr ; Kreiss, Heinz-Otto

University:California Institute of Technology

Department:Engineering and Applied Science

关键词: Oscillatory Differential Equations;   

Others  :  https://thesis.library.caltech.edu/1601/4/scheid_re_1982.pdf

美国|英语

来源: Caltech THESIS

【 摘 要 】

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)).