A numerical study of mean corrected phase-averaging in geophysical flow equations

This paper introduces a new algorithm to improve the accuracy of numerical phase-averaging in the oscillatory, multiscale, differential equations that govern geophysical fluid flow. Phase-averaging is a timestepping method that averages a mapped modulation variable in which the equations no longer contain a highly oscillatory linear term. This allows phase-averaging to retain the main contribution of fast waves on the low frequencies without explicitly resolving the rapid oscillations, thus allowing larger timesteps for explicit timesteppers. Whilst the modulation variable mapping is also used in operator splitting methods, and is accurate for asymptotic regimes, the advantage of phase-averaging is that low-frequency oscillations are retained, allowing for accuracy in systems further from the asymptotic limit. However, this comes at the cost of introducing an averaging error when computing phase-averages using a smooth kernel. This paper proposes a method to offset some averaging error through a modified mapping that includes a mean correction term to capture an average measure of nonlinear oscillation. We show that the mean corrected algorithm reduces phase-averaging errors in the swinging spring system and the one-dimensional rotating shallow water equations, which are an important geophysical fluid system for weather and climate applications. We discuss potential new directions for the method based on the outcome of the numerical experiments.