On the Mathematical Description of Time-Dependent Surface Water Waves

Düll, Wolf-Patrick

doi:10.1365/s13291-017-0173-6

Cited by 3 publications

(1 citation statement)

References 128 publications

(194 reference statements)

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“…Two important examples include the approximation of waves in the form of wave packets by the nonlinear Schrödinger (NLS) equation, and the approximation of long waves in shallow water by the Kortewegde Vries (KdV) equation. We refer the reader to [43,[142][143][144][145], the book [24] and the surveys [146,147] and references therein for more about reduced models and their mathematical justification. -Books and reviews.…”

Section: (C) Other Questions and Further Referencesmentioning

confidence: 99%

Recent advances on the global regularity for irrotational water waves

Ionescu¹,

Pusateri²

2017

Phil. Trans. R. Soc. A.

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Abstract. We review recent progress on the long-time regularity of solutions of the Cauchy problem for the water waves equations, in two and three dimensions.We begin by introducing the free boundary Euler equations and discussing the local existence of solutions using the paradifferential approach, as in [7,1,2]. We then describe in a unified framework, using the Eulerian formulation, global existence results for three dimensional and two dimensional gravity waves, see [70,146,145,87,5,6,79,80,136], and our joint result with Deng and Pausader [60] on global regularity for the 3D gravity-capillary model.We conclude this review with a short discussion about the formation of singularities, and give a few additional references to other interesting topics in the theory.

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Section: (C) Other Questions and Further Referencesmentioning

confidence: 99%

Recent advances on the global regularity for irrotational water waves

Ionescu¹,

Pusateri²

2017

Phil. Trans. R. Soc. A.

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Validity of the Nonlinear Schrödinger Approximation for the Two-Dimensional Water Wave Problem With and Without Surface Tension in the Arc Length Formulation

Düll

2020

Arch Rational Mech Anal

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On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model

Li¹,

Liu²,

Liu³

et al. 2017

Preprint

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There is a growing interest in investigating numerical approximations of the water wave equation in recent years, whereas the lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. In this work, we focus on a nonlocal hyperbolic model, which essentially inherits the features of the water wave equation, and is simplified from the latter. For the constant coefficient case, we carry out systematical stability studies of the fully discrete approximation of such systems with the Fourier spectral approximation in space and general Runge-Kutta method in time. In particular, we discover the optimal time step constraints, in the form of a modified CFL condition, when certain explicit Runge-Kutta method are applied. Besides, the convergence of the semi-discrete approximation of variable coefficient case is shown, which naturally connects to the water wave equation. Extensive numerical tests have been performed to verify the stability conditions and simulations of the simplified hyperbolic model in the high frequency regime and the water wave equation are also provided.

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On the Mathematical Description of Time-Dependent Surface Water Waves

Cited by 3 publications

References 128 publications

Recent advances on the global regularity for irrotational water waves

Recent advances on the global regularity for irrotational water waves

Validity of the Nonlinear Schrödinger Approximation for the Two-Dimensional Water Wave Problem With and Without Surface Tension in the Arc Length Formulation

On Runge-Kutta methods for the water wave equation and its simplified nonlocal hyperbolic model

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