A stroboscopic averaging algorithm for highly oscillatory delay problems

Abstract

We propose and analyze a heterogenous multiscale method for the efficient integration of constant-delay differential equations subject to fast periodic forcing. The stroboscopic averaging method (SAM) suggested here may provide approximations with O(H 2 + 1/Ω 2 ) errors with a computational effort that grows like H −1 (the inverse of the stepsize), uniformly in the forcing frequency Ω .

Word series high-order averaging of highly oscillatory differential equations with delay

“…To conclude let us point out that, for ordinary differential equations, it is possible to compute numerically a stroboscopically averaged solution Ξ without the explicit knowledge of the corresponding averaged system; the information on the averaged system required by the integrator is derived by numerically simulating the oscillatory system (1) [3,4]. Such techniques have been extended to delay differential equations [22,5].…”

“…In those cases finding high-order averaged systems may be extremely complicated (see e.g. the computations in [22]).…”

High-order stroboscopic averaging methods for highly oscillatory delay problems

“…the enhancement of the response to the slow forcing created by the presence of the high frequency forcing. For additional examples of problems of the form (1) see [24]. The application of standard software to the integration of (1) may be very expensive because accuracy typically requires that the step length be smaller than the small period T = 2π/Ω.…”

“…Here we follow the SAM technique [3,4] where the right hand-side of the averaged system is retrieved by using finite differences. A similar approach has been used in [24] but there are important differences between the algorithm in that reference and the integrators in this paper:…”

“…-The integrators suggested here are constructed by rewriting (1)- (2) as an ODE problem which is then solved by using the ODE SAM algorithms in [3,4]. Reference [24] borrows the main ideas of [3,4] and adjusts them to the delay problem (1)- (2). -A single algorithm is introduced in [24]; it is based on integrating the averaged problem with the second-order Adams-Basforth formula.…”