Automatic methods for highly oscillatory ordinary differential equations

Abstract

By a highly oscillatory ODE we mean one whose solution is “nearly periodic.” This paper is concerned with the low-cost, automatic detection of oscillatory behavior, the determination of its period, and methods for its subsequent efficient integration. In the first phase, the method for oscillatory problems discussed examines the output of an integrator to determine if the output is nearly periodic. At the point this answer is positive, the second phase is entered and an automatic multirevolutionary method is invoked to integrate a quasi-envelope of the solution. This requires the occasional solution of a nearly periodic initial-value problem over one period by a standard method and the re-determination of its period to provide the approximate derivatives of a quasi-envelope. The major difficulties addressed in this paper are the following: the determination of the point at which multirevolutionary methods are more economic, the automatic detection of stiffness in the multirevolutionary method (which uses a very large step), the calculation of the equivalent Jacobian for the multirevolutionary method (it is a transition matrix of the system over one period), and the calculation of a smooth quasi-envelope.

Supported in part by the Department of Energy, Contract DE-AC02-76ER02383.A003.