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An Efficient Numerical Method for Highly Oscillatory Ordinary Differential Equations

Linda R. Petzold
SIAM Journal on Numerical Analysis
Vol. 18, No. 3 (Jun., 1981), pp. 455-479
Stable URL: http://www.jstor.org/stable/2156866
Page Count: 25
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An Efficient Numerical Method for Highly Oscillatory Ordinary Differential Equations
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Abstract

A "quasi-envelope" of the solution of highly oscillatory differential equations is defined. For many problems this is a smooth function which can be integrated using much larger steps than are possible for the original problem. Since the definition of the quasi-envelope is a differential equation involving an integral of the original oscillatory problem, it is necessary to integrate the original problem over a cycle of the oscillation (to average the effects of a full cycle). This information can then be extrapolated over a long (giant!) time step. Unless the period is known a priori, it is also necessary to estimate it either early in the integration (if it is fixed) or periodically (if it is slowly varying). Error propagation properties of this technique are investigated, and an automatic program is presented. Numerical results indicate that this technique is much more efficient than conventional ODE methods, for many oscillating problems.

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