Iterated averaging of three-scale oscillatory systems

Abstract

A theory of iterated averaging is developed for a class of highly oscillatory ordinary differential equations (ODEs) with three well separated time scales. The solutions of these equations are assumed to be (almost) periodic in the fastest time scales. It is proved that the dynamics on the slowest time scale can be approximated by an effective ODE obtained by averaging out oscillations. In particular, the effective dynamics of the considered class of ODEs is always deterministic and does not show any stochastic effects. This is in contrast to systems in which the dynamics on the fastest time scale is mixing. The systems are studied from three perspectives: first, using the tools of averaging theory; second, by formal asymptotic expansions; and third, by averaging with respect to fast oscillations using nested convolutions with averaging kernels. The latter motivates a hierarchical numerical algorithm consisting of nested integrators.

The effect of two distinct fast time scales in the rotating, stratified Boussinesq equations: variations from quasi-geostrophy

“…The results of the previous Section yield a powerful description of the asymptotic behavior of the doubly singular limit described by (1). These results can be extended further than assuming an integer power (greater than 2) relationship between the small parameters and δ however.…”

“…The current investigation is of particular interest in a geophysical context when considered in light of the result of [41] that indicates that QG dominates the solution for sufficiently long times for solutions of the hydrostatic primitive equations. This relies on the fact that for this system (as in the case of the Boussinesq system considered in [14,43]) and the corresponding asymptotic reduction, the influence of the fast waves on the slow dynamics vanishes to O (1). We show that a similar condition holds for the reduced dynamics of the Boussinesq system when two distinct fast time scales are considered.…”

“…-As already established, we are concerned with cases for which L and K are skew-Hermitian with a discrete spectrum in which case their exponential is norm preserving and has purely imaginary, discrete spectrum. As noted in [13] these assumptions allow us to extend the theory of [34] to the nonlinear system (1). We note here that as shown in [13] these assumptions hold for the Boussinesq system considered in Section 5.…”

“…Though the PDE examples in this paper are from geophysical fluid dynamics, the approach outlined in Section 2 is generally applicable to any system (ordinary or partial differential equation) with two skew-Hermitian linear operators with a discrete spectrum, and a quadratic nonlinear term (provided sufficient conditions are placed on the strength of the nonlinearity, such as that discussed in [13]). As mentioned previously, the asymptotic reductions performed here are of interest in the context of time-parallel methods (see [15,2,20] and citations therein as well as [1] where three distinct time scales motivate a particular time-stepping algorithm).…”

Parareal Multiscale Methods for Highly Oscillatory Dynamical Systems

“…The main objective of many multiscale methods is efficient numerical approximations of ξ(u(t)) only. The general strategy of our algorithm is based on such multiscale methods for HiOsc ODEs that only resolve the macroscopic behavior of a system as specified by the slow variables [3,4,5,6,7,8,25,54]. In this respect, the algorithms listed above are different from other multiscale methods that resolve all scales of the dynamics, for example, multi-level methods or high-order asymptotic expansions [17,18,19,44,45].…”