Higher order phase averaging for highly oscillatory systems
File(s)
Accepted version
Author(s)
Bauer, Werner
Cotter, Colin
Wingate, Beth
Type
Journal Article
Abstract
We introduce a higher order phase averaging method for nonlinear oscillatory systems.Phase averaging is a technique to filter fast motions from the dynamics while still accounting fortheir effect on the slow dynamics. Phase averaging is useful for deriving reduced models that canbe solved numerically with more efficiency, since larger timesteps can be taken. Recently, Hautand Wingate [SIAM J. Sci. Comput., 36 (2014) pp. A693–A713] introduced the idea of computingfinite window numerical phase averages in parallel as the basis for a coarse propagator for a parallel-in-time algorithm. In this contribution, we provide a framework for higher order phase averagesthat aims to better approximate the unaveraged system while still filtering fast motions. While thebasic phase average assumes that the solution is independent of changes of phase, the higher ordermethod expands the phase dependency in a basis which the equations are projected onto. In this newframework, the original numerical phase averaging formulation arises as the lowest order version ofthis expansion in which the nonlinearity is projected onto the space of functions that are independentof the phase. Our new projection onto functions that arekth degree polynomials in the phase givesrise to higher order corrections to the phase averaging formulation. We illustrate the properties ofthis method on an ODE that describes the dynamics of a swinging spring due to Lynch (2002).Although idealized, this model shows an interesting analogy to geophysical flows as it exhibits a slowdynamics that arises through the resonance between fast oscillations. On this example, we showconvergence to the nonaveraged (exact) solution with increasing approximation order also for finiteaveraging windows. At zeroth order, our method coincides with a standard phase average, but athigher order it is more accurate in the sense that solutions of the phase averaged model track the solutions of the unaveraged equations more accurately.
Date Issued
2022-09
Date Acceptance
2022-03-02
Citation
SIAM: Multiscale Modeling and Simulation, 2022, 20 (3), pp.936-956
ISSN
1540-3459
Publisher
Society for Industrial and Applied Mathematics
Start Page
936
End Page
956
Journal / Book Title
SIAM: Multiscale Modeling and Simulation
Volume
20
Issue
3
Copyright Statement
© 2022 Society for Industrial and Applied Mathematics. Bauer, W., Cotter, C., & Wingate, B. (2022). Higher order phase averaging for highly oscillatory systems. Multiscale Modeling & Simulation, 20(3), 936-956.
License URL
Identifier
https://epubs.siam.org/doi/10.1137/21M1430546
Subjects
DYNAMICS
FLOWS
Mathematics
Mathematics, Interdisciplinary Applications
MULTISCALE METHODS
phase averaging
Physical Sciences
Physics
Physics, Mathematical
resonant interactions
Science & Technology
slow-fast systems
Publication Status
Published
Date Publish Online
2022-09-02