A mean correction for improved phase-averaging accuracy in oscillatory, multiscale, differential equations

Abstract

This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in oscillatory, multiscale, differential equations. Phase-averaging is a timestepping method which averages a mapped variable to remove highly oscillatory linear terms from the differential equation. This retains the main contribution of fast waves on the low frequencies without explicitly resolving the rapid oscillations. However, this comes at the cost of introducing an averaging error. To offset this, we propose a modified mapping that includes a mean correction term encoding an average measure of the nonlinear interactions. This mapping was introduced in Tao (2019) for weak nonlinearity and relied on classical time-averaging, which leaves only the zero frequencies. Our algorithm instead considers mean corrected phase-averaging when 1) the nonlinearity is not weak but the linear oscillations are fast and 2) finite averaging windows are applied via a smooth kernel, which has the advantage of retaining low frequencies whilst still eliminating the fastest oscillations. In particular, we introduce a local mean correction that combines the concepts of a mean correction and finite averaging; this retains low-frequency components in the mean correction that are removed with classical time-averaging. We show that the new timestepping algorithm reduces phase errors in the mapped variable for the swinging spring ODE in various dynamical configurations. We also show accuracy improvements with a local mean correction compared to standard phase-averaging in the one-dimensional rotating shallow water equations, a useful test case for weather and climate applications.

Figures (8)

• Figure 1: Errors for standard (solid line) and mean corrected (dashed) phase-averaging in the swinging spring experiment. The mean corrected method is more accurate for all the resonance factors examined, which models a range of in- and out-of-resonance interactions.
• Figure 2: Peddle plots of the errors over normalised averaging window, $\zeta = \eta /\Delta t$. In an out-of-resonance case (a), there is a clear minimum in the error for each method, with the best window for the mean corrected method being slightly larger. For the directly resonant case (b), the mean corrected method again is best with a larger window, but there are also multiple local minima in the error.
• Figure 3: Comparing the unaveraged modulation variable solution ($\mathbf{V}$, solid line), with the modulation variables of the standard phase-averaged method ($\mathbf{\overline{V}}$, dotted) and mean corrected method ($\mathbf{\overline{W}}$, dashed). This is in the directly resonant state of $\rho = 2$. The modulation variable solutions in (a) use the best $\zeta$ for the phase-averaged method, whilst (b) use the best $\zeta$ for the mean corrected method. The mean corrected method aligns better in the modulation space with its best averaging window in (b), whereas the phase-averaged method has considerable phase discrepancies, even in (a).
• Figure 4: Errors for the standard phase-averaged (solid line), classically mean corrected (dotted line), and locally mean corrected (dashed line) methods in the KG-type system. The two mean corrected methods are more accurate than the phase-averaged one for all values of $\epsilon$. The local mean correction has a lower error than the classical for $\epsilon \in \{ 0.1, 0.05 \}$. At $\epsilon=0.01$ there is minimal difference between $\mathbf{C}_{\infty}$ and $\mathbf{C}$, so both mean corrected methods perform similarly.
• Figure 5: Visualisations of the classical and local mean corrections in the KG-type system. The best $\eta_C$ for each method at $\Delta t = 1$ is used to compute mean corrections from a small timestep unaveraged modulation variable ($\mathbf{V}$) solution. The inverse Fourier transform is performed to give a physical (as opposed to spectral) space visualisation. Mean corrections from faster linear oscillations are shown moving down the page, with $\epsilon \in \{ 0.5, 0.1, 0.05, 0.01 \}$. The classical mean correction has a different profile for $\epsilon = 0.5$ but is visually indistinguishable in the other regimes. By contrast, the local mean correction has an observable variation between each $\epsilon$ regime. This is a result of the finite $\eta_C$ capturing different local mean dynamics depending on the oscillation frequency. At $\epsilon = 0.01$, the mean corrections are visually indistinguishable; this coincides with the two methods having similar accuracy.