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Rectification of a deep water model for surface gravity waves

Vincent Duchêne, Benjamin Melinand

Abstract

In this work we discuss an approximate model for the propagation of deep irrotational water waves, specifically the model obtained by keeping only quadratic nonlinearities in the water waves system under the Zakharov/Craig-Sulem formulation. We argue that the initial-value problem associated with this system is most likely ill-posed in finite regularity spaces, and that it explains the observation of spurious amplification of high-wavenumber modes in numerical simulations that were reported in the literature. This hypothesis has already been proposed by Ambrose, Bona, and Nicholls [4] but we identify a different instability mechanism. On the basis of this analysis, we show that the system can be "rectified". Indeed, by introducing appropriate regularizing operators, we can restore the well-posedness without sacrificing other desirable features such as a canonical Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient numerical integration. This provides a first rigorous justification for the common practice of applying filters in high-order spectral methods for the numerical approximation of surface gravity waves. While our study is restricted to a quadratic model, we believe it can be generalized to any order and paves the way towards the rigorous justification of a robust and efficient strategy to approximate water waves with arbitrary accuracy. Our study is supported by detailed and reproducible numerical simulations.

Rectification of a deep water model for surface gravity waves

Abstract

In this work we discuss an approximate model for the propagation of deep irrotational water waves, specifically the model obtained by keeping only quadratic nonlinearities in the water waves system under the Zakharov/Craig-Sulem formulation. We argue that the initial-value problem associated with this system is most likely ill-posed in finite regularity spaces, and that it explains the observation of spurious amplification of high-wavenumber modes in numerical simulations that were reported in the literature. This hypothesis has already been proposed by Ambrose, Bona, and Nicholls [4] but we identify a different instability mechanism. On the basis of this analysis, we show that the system can be "rectified". Indeed, by introducing appropriate regularizing operators, we can restore the well-posedness without sacrificing other desirable features such as a canonical Hamiltonian structure, cubic accuracy as an asymptotic model, and efficient numerical integration. This provides a first rigorous justification for the common practice of applying filters in high-order spectral methods for the numerical approximation of surface gravity waves. While our study is restricted to a quadratic model, we believe it can be generalized to any order and paves the way towards the rigorous justification of a robust and efficient strategy to approximate water waves with arbitrary accuracy. Our study is supported by detailed and reproducible numerical simulations.
Paper Structure (32 sections, 26 theorems, 262 equations, 8 figures)

This paper contains 32 sections, 26 theorems, 262 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. The equations and their rectification
  4. Description of the results and recommendations
  5. Discussion and prospects
  6. The instability mechanism
  7. A toy model
  8. Other models with quadratic precision
  9. The infinite-depth situation
  10. The shallow-water situation
  11. Higher order spectral method
  12. Outline
  13. Numerical study
  14. Numerical scheme
  15. Instabilities without rectification
  16. ...and 17 more sections

Key Result

Theorem 3.2

Let $d\in\{1,2\}$, $t_{0}> \frac{d}{2}$, $N \in \mathbb{N}$ with $N \geq t_{0}+2$, $C>1$ and $M>0$. Set $J\in L^\infty(\mathbb{R}^d)$ such that it defines regular rectifiers. There exists $T_0>0$ such that for any $\mu\geq 1$ and $\epsilon>0$, for any $(\zeta_{0},\psi_{0})\in H^N(\mathbb{R}^d)\times and for any $\delta \geq \epsilon M_0$, the following holds. There exists a unique $(\zeta,\psi)\i

Figures (8)

  • Figure 1: Time evolution of smooth initial data \ref{['num-init']}, without dealiasing and rectifiers.
  • Figure 2: Time evolution of smooth initial data \ref{['num-init']}, with dealiasing and without regularization.
  • Figure 3: Time evolution of smooth initial data \ref{['num-init']}, with a rectifier of order $m\in\{-1,-1/2\}$ .
  • Figure 4: Time evolution of smooth initial data \ref{['num-init']}, with a rectifier of order $m=-1/4$.
  • Figure 5: Time evolution of smooth initial data varying the strength of an admissible rectifier.
  • ...and 3 more figures

Theorems & Definitions (57)

  • Definition 1.1: Admissible rectifiers
  • Definition 3.1: Rectifiers
  • Theorem 3.2: Well-posedness
  • Theorem 3.3: Consistency
  • Theorem 3.4: Convergence
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Lemma 4.3
  • proof
  • ...and 47 more