Higher order phase averaging for highly oscillatory systems

by   Werner Bauer, et al.

We introduce a higher order phase averaging method for nonlinear oscillatory systems. Phase averaging is a technique to filter fast motions from the dynamics whilst still accounting for their effect on the slow dynamics. Phase averaging is useful for deriving reduced models that can be solved numerically with more efficiency, since larger timesteps can be taken. Recently, Haut and Wingate (2014) introduced the idea of computing finite 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 averages that aims to better approximate the unaveraged system whilst still filtering fast motions. Whilst the basic phase average assumes that the solution is independent of changes of phase, the higher order method expands the phase dependency in a basis which the equations are projected onto. We illustrate the properties of this 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 slow dynamics that arises through the resonance between fast oscillations. On this example, we show convergence to the non-averaged (exact) solution with increasing approximation order also for finite averaging windows. At zeroth order, our method coincides with a standard phase average, but at higher order it is more accurate in the sense that solutions of the phase averaged model track the solutions of the unaveraged equations more accurately.


page 14

page 17

page 18


Time parallel integration and phase averaging for the nonlinear shallow water equations on the sphere

We describe the application of the phase averaging technique to the nonl...

Optimized Geometric Constellation Shaping for Wiener Phase Noise Channels with Viterbi-Viterbi Carrier Phase Estimation

The Viterbi Viterbi (V V) algorithm is well understood for QPSK an...

Higher-Order Finite Element Approximation of the Dynamic Laplacian

The dynamic Laplace operator arises from extending problems of isoperime...

Semiexplicit Symplectic Integrators for Non-separable Hamiltonian Systems

We construct a symplectic integrator for non-separable Hamiltonian syste...

Randomized Kaczmarz with Averaging

The randomized Kaczmarz (RK) method is an iterative method for approxima...

Higher-order dependency pairs

Arts and Giesl proved that the termination of a first-order rewrite syst...

Improving numerical accuracy for the viscous-plastic formulation of sea ice

Accurate modeling of sea ice dynamics is critical for predicting environ...

Please sign up or login with your details

Forgot password? Click here to reset