The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint.

Esta dissertação tem por objetivo a implementação de um código para simular problemas hidrodinâmicos, bem como a possibilidade de controlar as elevações de onda resultantes numa determinada região por meio de uma vazão ótima controlada dentro do sistema estudado. O algoritmo implementado é baseado nas equações de águas rasas, as quais são aplicáveis em situações onde a altura d’água é de ordem muito menor do que as dimensões do sistema, que é discretizado espacial e temporalmente pelo Método dos Elementos Finitos e pelo método CBS (Characteristic Based-Split), respectivamente. O método de controle consiste na busca de uma curva de vazão de controle ótima que minimize a função objetivo, a qual compara os valores de altura d’água que se deseja encontrar em uma região especificada com os calculados pela simulação numérica. Para isso, utiliza-se um algoritmo evolutivo SCE-UA (Shuffled Complex Evolution – University of Arizona), que busca otimizar parâmetros de geração das curvas de vazão de controle, podendo estas serem modeladas por NURBS (Non- Uniform Rational B-Splines), que são capazes de encontrar a solução ótima, ou modeladas com curvas de forma triangular (linear) ou parabólica (quadrática) que apresentam uma solução aproximada de fácil implementação. Por fim, várias aplicações são realizadas, tanto para a simples simulação, quanto para o controle de problemas hidrodinâmicos, a fim de validar os algoritmos desenvolvidos e os resultados obtidos mostraram que os objetivos foram alcançados, encontrando uma forma eficiente de se fazer o controle de enchentes. / Implementation of a computational code for the numerical simulation of hydrodynamic problems as well as the ability to control the resulting wave elevations in a specific area, using an optimal flow controlled within the studied system are the aims of this work. The implemented algorithm is based on the shallow waters equations, which are applicable in situations where the water height is much smaller than the system dimensions, and are spatial and temporally discretized by the Finite Element Method and the CBS method (Caractheristic Based-Split), respectively. The control method consists in finding an optimal control flow curve that minimizes the objective function, which compares the objective value of water elevations in a specified region with those calculated by numerical simulation. An evolutionary algorithm called SCE-UA (Shuffled Complex Evolution - University of Arizona), which looks for optimize parameters of control flow curves generation, is used. These curves may be modeled by NURBS (Non-Uniform Rational B-Splines) which are able to find the optimal solution, or by curves of triangular (linear) or parabolic quadratic forms, which are an approximate solution easy to implement. Finally, several applications are performed for both simulation and control of hydrodynamic problems in order to validate the developed algorithms, and the results showed that the aims of this work were reached, finding an efficient way to control floods.

Esta dissertação tem por objetivo a implementação de um código para simular problemas hidrodinâmicos, bem como a possibilidade de controlar as elevações de onda resultantes numa determinada região por meio de uma vazão ótima controlada dentro do sistema estudado. O algoritmo implementado é baseado nas equações de águas rasas, as quais são aplicáveis em situações onde a altura d’água é de ordem muito menor do que as dimensões do sistema, que é discretizado espacial e temporalmente pelo Método dos Elementos Finitos e pelo método CBS (Characteristic Based-Split), respectivamente. O método de controle consiste na busca de uma curva de vazão de controle ótima que minimize a função objetivo, a qual compara os valores de altura d’água que se deseja encontrar em uma região especificada com os calculados pela simulação numérica. Para isso, utiliza-se um algoritmo evolutivo SCE-UA (Shuffled Complex Evolution – University of Arizona), que busca otimizar parâmetros de geração das curvas de vazão de controle, podendo estas serem modeladas por NURBS (Non- Uniform Rational B-Splines), que são capazes de encontrar a solução ótima, ou modeladas com curvas de forma triangular (linear) ou parabólica (quadrática) que apresentam uma solução aproximada de fácil implementação. Por fim, várias aplicações são realizadas, tanto para a simples simulação, quanto para o controle de problemas hidrodinâmicos, a fim de validar os algoritmos desenvolvidos e os resultados obtidos mostraram que os objetivos foram alcançados, encontrando uma forma eficiente de se fazer o controle de enchentes. / Implementation of a computational code for the numerical simulation of hydrodynamic problems as well as the ability to control the resulting wave elevations in a specific area, using an optimal flow controlled within the studied system are the aims of this work. The implemented algorithm is based on the shallow waters equations, which are applicable in situations where the water height is much smaller than the system dimensions, and are spatial and temporally discretized by the Finite Element Method and the CBS method (Caractheristic Based-Split), respectively. The control method consists in finding an optimal control flow curve that minimizes the objective function, which compares the objective value of water elevations in a specified region with those calculated by numerical simulation. An evolutionary algorithm called SCE-UA (Shuffled Complex Evolution - University of Arizona), which looks for optimize parameters of control flow curves generation, is used. These curves may be modeled by NURBS (Non-Uniform Rational B-Splines) which are able to find the optimal solution, or by curves of triangular (linear) or parabolic quadratic forms, which are an approximate solution easy to implement. Finally, several applications are performed for both simulation and control of hydrodynamic problems in order to validate the developed algorithms, and the results showed that the aims of this work were reached, finding an efficient way to control floods.

The thesis deals with the numerical solution of partial differential equati- ons describing the flow of the so-called shallow water neglecting the flow in the vertical direction. These equations are of hyperbolical type of the first or- der with a reactive term representing the bottom topology. We discretize the resulting system of equations by the implicit space-time discontinuous Ga- lerkin method (STDGM). In the literature, the explicit techniques are used most of the time. The implicit approach is suitable especially for adaptive methods, because it allows the usage of different meshes for different time niveaus. In the thesis we derive the corresponding method and an adaptive algorithm. Finally, we present usage of the method in several examples. 1

Nous nous intéressons à l’étude théorique et numérique de plusieurs modèles d’écoulement (Saint-Venant, multicouches, milieux poreux stationnaires et non stationnaires) et de leurs applications à l’optimisation de formes de certains ouvrages hydrauliques. Nous explorons le caractère bien posé des systèmes, nous dérivons un système adjoint lié à chaque modèle.Une méthode de pénalisation est utilisée pour relaxer la contrainte d’incompressibilité de la vitesse.Nous exprimons le gradient de forme en fonction de la vitesse u comme variable d’état, des variables adjointes, et le vecteur unité normal au bord du domaine.Nous adoptons une méthode d’éléments finis discrète pour approcher la solution du problème pénalisé et établissons des estimations à priori afin de prouver la convergence de la solution approchée vers la solution du système non perturbé.Le problème d’optimisation est implémenté en utilisant la méthode adjointe continue et la méthode d’éléments finis. / We are interested in the theoretical and numerical study of different flow models (shallow water system, multilayer, stationary and non stationary porous media) and their applications to the shape optimization of some hydraulic structures.We explore the well-posedness of the models and derive the adjoint equations related to each system.A penalty method is used to relax the incompressibility constraint for the velocity. We express the shape gradient of the cost function in terms of the velocity value as a state variable, the adjoint variables and the unit normal vector to the boundary of the domain.We propose a discrete finite element method to approximate the solution for the penalizedproblem and establish a priori estimates to prove the convergence of the approximate solution to the solution of the non perturbed problem. Error estimates for the velocity and the pressure are established.The optimization procedure is implemented using the continuous adjoint method and the finite element method.

Flooding has always been and is still today a disastrous event with agricultural, infrastructural, economical and not least humanitarian ramifications. Understanding the behaviour of floods is crucial to be able to prevent or mitigate future catastrophes, a task which can be accomplished by modelling the water flow. In this thesis the finite element method is employed to solve the shallow water equations, which govern water flow in shallow environments such as rivers, lakes and dams, a methodology that has been widely used for flooding simulations. Alternative approaches to model floods are however also briefly discussed. Since the finite element method suffers from numerical instabilities when solving nonlinear conservation laws, the shallow water equations are stabilised by introducing a high-order nonlinear artificial viscosity, constructed using a multi-mesh strategy. The accuracy, robustness and well-balancedness of the solution are examined through a variety of benchmark tests. Finally, the equations are extended to include a friction term, after which the effectiveness of the method in a real-life scenario is verified by a prolonged simulation of the Malpasset dam break.

Observations of Jupiter show a super-rotating (prograde) equatorial jet that has persisted for decades. Shallow water simulations run in the Jovian parameter regime reproduce the mixture of robust vortices and alternating zonal jets observed on Jupiter, but the equatorial jet is invariably sub-rotating (retrograde). Recent work has obtained super-rotating equatorial jets by extending the standard shallow water equations to relax the height field towards its mean value. This Newtonian cooling-like term is intended to model radiative cooling to space, but its addition breaks key conservation properties for mass and momentum. In this thesis the radiatively damped thermal shallow water equations are proposed as an alternative model for Jovian atmospheres. They extend standard shallow water theory by permitting horizontal variations of the thermodynamic properties of the fluid. The additional temperature equation allows a Newtonian cooling term to be included while conserving mass and momentum. Simulations reproduce equatorial jets in the correct directions for both Jupiter and Neptune (which sub-rotates). Quasi-geostrophic theory filters out rapidly moving inertia-gravity waves. A local quasi-geostrophic theory of the radiatively damped thermal shallow water equations is derived, and then extended to cover whole planets. Simulations of this global thermal quasi-geostrophic theory show the same transition, from sub- to super-rotating equatorial jets, seen in simulations of the original thermal shallow water model as the radiative time scale is decreased. Thus the mechanism responsible for setting the direction of the equatorial jet must exist within quasi-geostrophic theory. Such a mechanism is developed by calculating the competing effects of Newtonian cooling and Rayleigh friction upon the zonal mean zonal acceleration induced by equatorially trapped Rossby waves. These waves transport no momentum in the absence of dissipation. Dissipation by Newtonian cooling creates an eastward zonal mean zonal acceleration, consistent with the formation of super-rotating equatorial jets in simulations, while the corresponding acceleration is westward for dissipation by Rayleigh friction.

Flood inundation is a major natural hazard that can have very severe socio-economic consequences. This thesis presents an enhanced numerical model for flood simulation. After setting the context by examining recent large-scale flood events, a literature review is provided on shallow flow numerical models. A new version of the hyperbolic horizontal variable density shallow water equations with source terms in balanced form is used, designed for flows over complicated terrains, suitable for wetting and drying fronts and erodible bed problems. Bed morphodynamics are included in the model by solving a conservation of bed mass equation in conjunction with the variable density shallow water equations. The resulting numerical scheme is based on a Godunov-type finite volume HLLC approximate Riemann solver combined with MUSCL-Hancock time integration and a non-linear slope limiter and is shock-capturing. The model can simulate trans-critical, steep-fronted flows, connecting bodies of water at different elevations. The model is validated for constant density shallow flows using idealised benchmark tests, such as unidirectional and circular dam breaks, damped sloshing in a parabolic tank, dam break flow over a triangular obstacle, and dam break flow over three islands. The simulation results are in excellent agreement with available analytical solutions, alternative numerical predictions, and experimental data. The model is also validated for variable density shallow flows, and a parameter study is undertaken to examine the effects of different density ratios of two adjacent liquids and different hydraulic thrust ratios of species and liquid in mixed flows. The results confirm the ability of the model to simulate shallow water-sediment flows that are of horizontally variable density, while being intensely mixed in the vertical direction. Further validation is undertaken for certain erodible bed cases, including deposition and entrainment of dilute suspended sediment in a flat-bottomed tank with intense mixing, and the results compared against semi-analytical solutions derived by the author. To demonstrate the effectiveness of the model in simulating a complicated variable density shallow flow, the validated numerical model is used to simulate a partial dam-breach flow in an erodible channel. The calibrated model predictions are very similar to experimental data from tests carried out at Tsinghua University. It is believed that the present numerical solver could be useful at describing local horizontal density gradients in sediment laden and debris flows that characterise certain extreme flood events, where sediment deposition is important.

In the present work we study the numerical solution of shallow water equations. We introduce a vectorial notation of equations laws of conservation from which we derive the shallow water equations (SWE). There is the simplify its derivation, notation and the most important features. The original contribution is to derive equations for shallow water without the using of Leibniz's formula. There we report the finite volume method with the numerical flow of Vijayasundaram type for SWE. We present a description of the linear reconstruction, quadratic reconstruction and ENO reconstruction and their using for increasing of order accuracy. We demonstrate using of linear reconstruction in finite volume method of second order accuracy. This method is programmed in Octave language and used for solving of two problems. We apply the method of the ADER type for the shallow water equations. This method was originally designed for the Euler's equation.

In der vorliegenden Arbeit wird das Entstehen von Singularitäten an Oberflächen von dünnen Flüssigkeitsfilmen studiert. Unter einer Singularität versteht man hier das plötzliche Aufreißen einer Flüssigkeitsoberfläche an einer Stelle. Nach einer Diskussion physikalischer Phänomene, wird ein 2D Modell zur Beschreibung von Flüssigkeitsfilmen hergeleitet. Dieses Modell beinhaltet u.a. Oberflächenspannung, van der Waals''sche Kräfte und eine Navier-slip Randbedingung (Schlupf-Randbedingung) zwischen Substrat und Flüssigkeit, d.h. die Flüssigkeite haftet nicht an der Grenzfläche zum Substrat. Dieses Phänomen wird vor allen Dingen im Nano- und Mikrometerbereich beobachtet. Dieses Modell wird vereinfacht und man erhält die sogenannte "strong-slip" Gleichung. In dieser Dissertation werden verschiedene Ansätze verfolgt, um die Singularität der Flüssigkeitsoberfläche zu beschreiben. Der Entstehungsprozess der Singularität wird durch die lineare Stabilitätsuntersuchung beschrieben. Da die Linearisierung schnell ihre Gültigkeit verliert, wird das nichtlineare Verhalten der Singularität mit einem numerischen Verfahren beschrieben. Das dazu hier konstruierte Finite-Differenzen-Schema besitzt eine hohe räumliche und zeitliche Genauigkeit. Dadurch können verschiedene Regime, in denen die Singularität eine selbstähnliche Dynamik besitzt, untersucht und beschrieben werden. Im zweiten Teil der Arbeit werden die Gleichungen weiter vereinfacht. Dadurch können qualitative Eigenschaften der Singularitätsentstehung bewiesen werden. Weiterhin kann so eine Verbindung zu Modellen der Ostwald-Reifung hergestellt werden und man gelangt zu ähnlichen mathematischen Aussagen wie für selbstähnliche Vergröberungsprozesse. Insbesondere wird in der Arbeit gezeigt, dass die Singularität nach endlicher Zeit auftritt. Für das vereinfachte Problem werden hinreichende und notwendige Bedingungen für selbstähnliches Verhalten angegeben. / In this thesis we study the formation of surface singularites of thin liquid films, i.e., rupture of thin liquid films. First, important physical phenomena are discussed and a two-dimensional model for thin-film rupture is derived . That model contains surface tension, van der Waals forces between a liquid and a underlying substrate, and a Navier-slip condition. Using the thin-film hypothesis, this model is simplified and one obtains the so-called strong-slip equation. The phenomenon slip, where the velocity of the liquid is non-zero at a fluid-solid interface, is particularly important at microscopic length scales. In this text we study interfacial singularities with various approaches. The creation of a singularity is described by a linear stability analysis. The non-linear behavior is investigated by a numerical analysis. A finite-difference scheme is used to study the non-linear self-similar dynamics of the singularity. In the second part of this thesis the equations are further simplified. This allows to study qualitative properties of the singularity formation. Furthermore, we can establish a correspondence to models for Ostwald rippending and obtain similar mathematical statements as they are known for self-similar coarsening processes. In particular it is shown that rupture happens after a finite time. In addition, necessary and sufficient condition for self-similar rupture are proven.