High-order stroboscopic averaging methods for highly oscillatory delay problems

Abstract

We introduce and analyze a family of heterogeneous multiscale methods for the numerical integration of highly oscillatory systems of delay differential equations with constant delays. The methodology suggested provides algorithms of arbitrarily high accuracy.Mathematical Subject Classification (2010) 65L03, 34C29 Keywords Delay differential equations, stroboscopic averaging, highly oscillatory problems IntroductionThis paper suggests and analyzes heterogeneous multiscale methods [10,11,13,12,18,1,23,5] for the numerical solution of highly oscillatory systems of delay differential equations (DDEs) with constant delays. The methods may achieve arbitrarily high orders of convergence and are based on the idea of the stroboscopic averaging method (SAM) [3,4] for highly oscillatory ordinary differential equations (ODEs).

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 this way, the time-dependent change of variables ξ → Ξ in (4) transforms the given highly oscillatory initial value problem into the initial value problem (5) where there is no periodic forcing. Therefore (5) provides an averaged version of (1). In addition, the coefficients κ w (θ) are such that, at θ = 2kπ, k = 0, 1, 2, .…”

“…That x(t) − X(t) = O(1/Ω) at stroboscopic times t ≤ τ follows by considering that, for those values of t, x(t) = x (1) (t), X(t) = X (1) (t) and x (1) (t) − X (1) (t) = O(1/Ω) because of the properties of stroboscopic averaging of ordinary differential equations without delay. For stroboscopic times t in the interval τ < t ≤ 2τ the situation is slightly more complicated because in addition to the error O(1/Ω) arising from truncating (5) there is an additional source of error: in this interval x(t) = x (2) (t + τ ), X(t) = X (2) (t + τ ) but x (2) (t) and X (2) (t) do not share the same initial value at t = 0. However the difference of initial values is x (2) (0) − X (2) (0) = x (1) (τ ) − X (1) (τ ), which we know is of size O(1/Ω).…”