On WW2

Rectifying a deep water waves model

Vincent Duchêne
Mar 7, 2022 3 min read

Together with Benjamin Melinand, we have uploaded an arXiv preprint, which is the conclusion of many discussions over the past years (some including Jean-Claude Saut), and hopefully the starting point of a longer-term project.

In this work we study the so-called “WW2” model, which is the first (nonlinear) system of a hierarchy of models put advocated by Craig and Sulem1 as a mean for efficiently producing approximate numerical solutions to the water waves system.2 These models have nice features (including a canonical Hamiltonian structure) and are perfectly suited to pseudo-spectral numerical methods.3 Yet when implementing this approach,4 it was observed the emergence of high-frequency instabilities, which are typically suppressed through ad-hoc regularizations such as parabolic regularization or low-pass filters. Later on, Ambrose, Bona and Nicholls5 suggested that these instabilities are not spurious effects from the numerical discretization of the equations, but rather the consequence of the ill-posedness of the equations at stake.

Two images showing the result of a numerical simulation at three different times, both in physical and Fourier variables (the latter in log scale)
The WW2 system exhibits high-frequency instabilities.

With Benjamin Melinand, we agree with the conclusion of Ambrose, Bona and Nicholls, yet not on the instability mechanism that they suggest. Through a detailed investigation of the high-frequency component of the equations, we show that the quadratic WW2 model produces cubic instabilities, fairly similar to Kelvin-Helmholtz instabilities.6 This allows us to propose a “rectification” procedure: by inserting regularizing operators with the correct scaling, we can recover suitable stability properties without damaging the precision of the model. This allows us to fully justify the (rectified) WW2 system as model for water waves in the deep water regime.

Two images showing the result of a numerical simulation at two different times, both in physical and Fourier variables (the latter in log scale)
The instabilities can be suppressed, without harming the precision of the model.

Our analysis is supported by an in-depth numerical investigation which is fully reproducible, thanks to the Julia package ‘WaterWaves1D.jl’ which we have developped with Pierre Navaro.

Among the natural perspectives emerging from this work, one can ask whether the analysis can be applied to the full hierarchy of models mentioned above. We believe it does, and that it is possible to “rectify” all of the models in such a way that the rectified hierarchy will produce a converging sequence of approximate solutions. Thi is a work in progress.

  1. W. Craig and C. Sulem, Numerical simulation of gravity waves. J. Comput. Phys. , Vol. 108, No. 1:73–83, 1993 ↩︎

  2. describing the motion of a layer of homogeneous incompressible fluid with a free surface, under the influence of gravity, in the irrotational framework. See e.g. my memoir. ↩︎

  3. described here. ↩︎

  4. P. Guyenne and D. P. Nicholls, A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM J. Sci. Computer, no. 1:81–101, 2007-08 ↩︎

  5. D. M. Ambrose, J. L. Bona, and D. P. Nicholls, On ill-posedness of truncated series models for water waves. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 470(2166):20130849, 16, 2014 ↩︎

  6. described here. Notice however, that the Kelvin-Helmholtz instabilities are physically relevant while the instabilities in WW2 are spurious effects of the modeling procedure. ↩︎

Vincent Duchêne
Vincent Duchêne
Chargé de Recherche CNRS