On surface water waves and tsunami propagation

In dieser Arbeit werden die reibungslosen Bewegungsgleichungen für wasser Wellen mit physikalischer Motivation eingeführt. Es folgt ein Studium der Eigenschaften dieser Gleichungen, die durch anwendung asymptotischer Näherungen zur Korteweg-de Vries Gleichung führen. Schließlich wird die Korteweg-de Vries Gleichung hinsichtlich ihrer Anwendung im Bereich der Tsunami Modellierung untersucht.This work introduces the inviscid governing equations for water waves from a physically motivated standpoint, in as accessible a manner as possible. From there, certain asymptotic regimes are explored, leading to the Korteweg-de Vries equation. Elaborations are made on applications to tsunami modeling, while taking care to point out shortcomings in the analytical approach as well as unresolved difficulties in reconciling the intriguing nature of water with mathematics

Nonlinear Waves and Dispersive Equations

The aim of the workshop was to discuss current developments in nonlinear waves and dispersive equations from a PDE based view. The talks centered around rough initial data, long time and global existence, perturbations of special solutions, and applications

Météotsunamis, résonance de Proudman et effet Coriolis pour les équations de vagues

In this work, we are interested in the evolution of water waves under the influence of a non constant atmospheric pressure, a moving bottom and a Coriolis forcing. In a first part, we study the Proudman resonance. We propose a mathematical approach to understand this phenomenon. First, we prove a local wellposedness result in a irrotational framework on the water waves equations (also called the Zakharov/Craig-Sulem formulation). Then, we fully justify different asymptotic models. In particular, we carefully study the Proudman resonance in deep water in the linear regime. Finally, we study the propagation of water waves in a weakly nonlinear regime thanks to the Saut-Xu equations and we propose a numerical scheme in order to solve these equations. In a second part, we study the influence of a Coriolis forcing on water waves. We prove a local wellposedness result on the Castro-Lannes equations, which generalize the Zakharov/Craig-Sulem formulation in the rotational framework. Then, we fully justify different asymptotic models when we take into account a Coriolis forcing. In particular, we generalize the Boussinesq equations (asymptotic model in a weakly nonlinear regime) in this setting. Thanks to these equations, we justify the Poincaré waves and then the Ostrovsky equation, which generalize the Korteweg-De- Vries equation when a Coriolis forcing is taking into account.Dans ce travail nous nous intéressons aux comportement de vagues soumises à l’action d’une pression atmosphérique non constante, un fond mobile et la force de Coriolis. Une première partie est dédiée à l’étude de la résonance de Proudman. Nous proposons une approche mathématique rigoureuse pour étudier ce phénomène. Nous commençons par démontrer un résultat d’existence locale dans un cadre irrotationnel sur les équations des vagues (appelées aussi formulation de Zakharov/Craig-Sulem). Puis, nous justifions différents modèles asymptotiques pour généraliser cette résonance dans diverses situations physiques. Nous proposons en particulier une étude détaillée dans des eaux profondes dans un régime linéaire. Nous étudions aussi la propagation de vagues dans des eaux profondes dans un régime faiblement non-linéaire grâce aux équations de Saut-Xu et nous proposons un schéma numérique pour résoudre ces équations. Dans une deuxième partie, nous étudions l’effet de la force de Coriolis sur les vagues. Nous démontrons un résultat d’existence locale sur les équations Castro-Lannes, équations qui généralisent la formulation de Zakharov/Craig-Sulem dans un cadre rotationnel. Nous justifions ensuite différents modèles asymptotiques dans des eaux peu profondes en présence de la force de Coriolis. En particulier, nous proposons une généralisation des équations de Boussinesq (modèle asymptotique dans un régime faiblement linéaire) lorsque la force de Coriolis n’est pas négligeable. Ces équations nous permettent ensuite de justifier mathématiquement les ondes de Poincaré puis l’équation d’Ostrovsky qui généralise l’équation de Korteweg-De-Vries en présence de la force de Coriolis

A hybridized discontinuous Galerkin method for nonlinear dispersive water waves

Simulation of water waves near the coast is an important problem in different branches of engineering and mathematics. For mathematical models to be valid in this region, they should include nonlinear and dispersive properties of the corresponding waves. Here, we study the numerical solution to three equations for modeling coastal water waves using the hybridized discontinuous Galerkin method (HDG). HDG is known to be a more efficient and in certain cases a more accurate alternative to some other discontinuous Galerkin methods, such as local DG. The first equation that we solve here is the Korteweg-de Vries equation. Similar to common HDG implementations, we first express the approximate variables and numerical fluxes in each element in terms of the approximate traces of the scalar variable, and its first derivative. These traces are assumed to be single-valued on each face. We next impose the conservation of the numerical fluxes via two sets of equations on the element boundaries. We solve this equation by Newton-Raphson method. We prove the stability of the proposed method for a proper choice of stabilization parameters. Through numerical examples, we observe that for a mesh with kth order elements, the computed variable and its first and second derivatives show optimal convergence at order k + 1 in both linear and nonlinear cases, which improves upon previously employed techniques. Next, we consider solving the fully nonlinear irrotational Green-Naghdi equation. This equation is often used to simulate water waves close to the shore, where there are significant dispersive and nonlinear effects involved. To solve this equation, we use an operator splitting method to decompose the problem into a dispersive part and a hyperbolic part. The dispersive part involves an implicit step, which has regularizing effects on the solution of the problem. On the other hand, for the hyperbolic sub-problem, we use an explicit hybridized DG method. Unlike the more common implicit version of the HDG, here we start by solving the flux conservation condition for the numerical traces. Afterwards, we use these traces in the original PDEs to obtain the internal unknowns. This process involves Newton iterations at each time step for computing the numerical traces. Next, we couple this solver with the dispersive solver to obtain the solution to the Green-Naghdi equation. We then solve a set of numerical examples to verify and validate the employed technique. In the first example we show the convergence properties of the numerical method. Next, we compare our results with a set of experimental data for nonlinear water waves in different situations. We observe close to optimal convergence rates and a good agreement between our numerical results and the experimental data.Engineering Mechanic

Nonsemilinear one-dimensional PDEs: analysis of PT deformed models and numerical study of compactons

This thesis is based on the work done during my PhD studies and is roughly divided in two independent parts. The first part consists of Chapters 1 and 2 and is based on the two papers Cavaglià et al. [2011] and Cavaglià & Fring [2012], concerning the complex PT-symmetric deformations of the KdV equation and of the inviscid Burgers equation, respectively. The second part of the thesis, comprising Chapters 3 and 4, contains a review and original numerical studies on the properties of certain quasilinear dispersive PDEs in one dimension with compacton solutions. The subjects treated in the two parts of this work are quite different, however a common theme, emphasised in the title of the thesis, is the occurrence of nonsemilinear PDEs. Such equations are characterised by the fact that the highest derivative enters the equation in a nonlinear fashion, and arise in the modeling of strongly nonlinear natural phenomena such as the breaking of waves, the formation of shocks and crests or the creation of liquid drops. Typically, nonsemilinear equations are associated to the development of singularities and non-analytic solutions. Many of the complex deformations considered in the first two chapters are nonsemilinear as a result of the PT deformation. This is also a crucial feature of the compacton-supporting equations considered in the second part of this work. This thesis is organized as follows. Chapter 1 contains an introduction to the field of PT-symmetric quantum and classical mechanics, motivating the study of PT-symmetric deformations of classical systems. Then, we review the contents of Cavaglià et al. [2011] where we explore travelling waves in two family of complex models obtained as PT-symmetric deformations of the KdV equation. We also illustrate with many examples the connection between the periodicity of orbits and their invariance under PTsymmetry. Chapter 2 is based on the paper Cavaglià & Fring [2012] on the PTsymmetric deformation of the inviscid Burgers equation introduced in Bender & Feinberg [2008]. The main original contribution of this chapter is to characterise precisely how the deformation affects the gradient catastrophe. We also point out some incorrect conclusions of the paper Bender & Feinberg [2008]. Chapter 3 contains a review on the properties of nonsemilinear dispersive PDEs in one space dimension, concentrating on the compacton solutions discovered in Rosenau & Hyman [1993]. After an introduction, we present some original numerical studies on the K(2, 2) and K(4, 4) equations. The emphasis is on illustrating the different type of phenomena exhibited by the solutions to these models. These numerical experiments confirm previous results on the properties of compacton-compacton collisions. Besides, we make some original observations, showing the development of a singularity in an initially smooth solution. In Chapter 4 , we consider an integrable compacton equation introduced by Rosenau in Rosenau [1996]. This equation has been previously studied numerically in an unpublished work by Hyman and Rosenau cited in Rosenau [2006]. We present an independent numerical study, confirming the claim of Rosenau [2006] that travelling compacton equations to this equation do not contribute to the initial value problem. Besides, we analyse the local conservation laws of this equation and show that most of them are violated by any solution having a compact, dynamically evolving support. We confirm numerically that such solutions, which had not been described before, do indeed exist. Finally, in Chapter 5 we present our conclusions and discuss open problems related to this work