We investigate the role of boundary layer forcing and surface heterogeneities on the intensity and spectral distribution of the convective circulations of an idealized convective system. Our ultimate goal is to further the understanding of atmospheric convection. However, we depart from realistic atmospheric convection and study an idealized convective system known as the Rayleigh-Benard model in two dimensions. We extended the classical Rayleigh-Benard model to include the effects of boundary heterogeneities. These effects are included, inparticular through a sinusoidally variable surface temperature. In this idealized model, the Rayleigh number plays the role of convective available potential energy (CAPE) in atmospheric convection, while the boundary heterogeneities in the temperatureplay the role of boundary layer forcing. In particular, we study the effects of boundary forcing on the intensity and spectral distribution of convective circulations in great detail.We consider the problem in the linear and weakly nonlinear regimes. In the linear regime, we find an analytical solution for Rayleigh-Benard convection with boundary forcing. We show that the inclusion of periodic boundary forcing causes discontinuities in the linear solution when critical conditions are approached. In the nonlinear regime, we find the solution by direct numerical simulation. The nonlinearities not only remove the discontinuities, but also lead to the appearance of non-trivial modes in the solution.The classical modes appear when the Rayleigh number issupercritical and the amplitude of the boundary forcing is small. Modes governed by boundary forcing dominate when its amplitude is large. Non-trivial modes with wavenumbers different from either the classical or the boundary modes appear only for intermediate values of the boundary forcing. The transitions between regions dominated by the classical Rayleigh forcing, mixed forcing, andboundary forcing depend on the Rayleigh number and the wavenumber of the boundary forcing. We conclude that boundary forcing has non-trivial effects on convective circulations. This result might have important implications for atmosphericconvection. Indeed, it suggests that atmospheric convection over the relatively homogeneous oceans would have different spectral distribution compared to that over heterogeneous land surfaces. This result is consistent with observations.

Title: Numerical analysis of approximation of nonpolygonal domains for discon- tinuous Galerkin method Author: Filip Klouda Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc., KNM MFF UK Abstract: In this work we use the discontinuous Galerkin finite element method for the semidiscretization of a nonlinear nonstationary convection-diffusion pro- blem defined on a nonpolygonal two-dimensional domain. Using so called appro- ximating curved elements we define a piecewise polynomial approximation of the boundary of the domain and a space on which we search for a solution. We study the convergence of the method considering a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The obtained results allow us to derive an error estimate for the Discontinuous Galer- kin method employing the approximating curved elements. This estimate depends on the order of the approximation of the solution and also on the order of the approximation of the boundary. We describe one possibility of the construction of the approximating curved elements with the aid of a polynomial mapping given by an interpolation of points on the boundary. We present numerical experiments. Keywords: nonlinear convection-diffusion equation, discontinuous...

In the present work we deal with the stability of the space-time discontinuous Galerkin method applied to non-stationary, nonlinear convection - diffusion problems. Discontinuous Galerkin method is a very efficient tool for numerical solution of partial differential equations, combines the advantages of the finite element method (polynomial approximations of high order of accuracy) and the finite volume method (discontinuous approximations). After the formulation of the continuous problem its discretization in space and time is described. In the formulation of the discontinuous Galerkin method the non-symmetric, symmetric and incomplete version of discretization of the diffusion term is used and there are added penalty terms to the scheme also. In the third chapter are estimated individual terms of the previously derived approximate solution by special norms. Using the concept of discrete characteristic functions and the discrete Gronwall lemma, it is shown that the analyzed scheme is unconditionally stable. At the end, in the fourth chapter, are given some numerical experiments, which verify theoretical results from the previous chapter.

This thesis is concerned with analysis and implementation of Time discontinuous Galerkin method. Important part of it is constructing of algorithm for solving nonlinear convection-diffusion equations, which combines Discontinuous Galerkin method in space (DGFEM) with Time discontinuous Galerkin method (TDG). Nonlinearity of the problem is overcome by damped Newton-like method. This approach provides easy adaptivity manipulation as well as high order approximation with respect to both space and time variables. The second part of the thesis is focused on Time discontinuous Galerkin method, applied to ordinary differential equations. It is shown that the solution of Time discontinuous Galerkin equals the solution obtained by Radau IIA implicit Runge-Kutta method in the roots of right Radau Quadrature. By virtue of this relation, error estimates of the order higher by one than the standard order can be obtained in these points. Furthermore, almost two times higher order can be achieved in the endpoints of the intervals of time discretization. Finally, the thesis deals with the phenomenon of stiffness, which may dramatically decrease the order of the applied method. The theoretical results are verified by numerical experiments. Powered by TCPDF (www.tcpdf.org)

Cette thèse porte sur la modélisation de l’écoulement et du transport en milieu poreux ;nous effectuons des simulations numériques et démontrons des résultats de convergence d’algorithmes.Au Chapitre 1, nous appliquons des méthodes de volumes finis pour la simulation d’écoulements à densité variable en milieu poreux ; il vient à résoudre une équation de convection diffusion parabolique pour la concentration couplée à une équation elliptique en pression.Nous nous appuyons sur la méthode des volumes finis standard pour le calcul des solutions de deux problèmes spécifiques : une interface en rotation entre eau salée et eau douce et le problème de Henry. Nous appliquons ensuite la méthode de volumes finis généralisés SUSHI pour la simulation des mêmes problèmes ainsi que celle d’un problème de bassin salé en dimension trois d’espace. Nous nous appuyons sur des maillages adaptatifs, basés sur des éléments de volume carrés ou cubiques.Au Chapitre 2, nous nous appuyons de nouveau sur la méthode de volumes finis généralisés SUSHI pour la discrétisation de l’équation de Richards, une équation elliptique parabolique pour le calcul d’écoulements en milieu poreux. Le terme de diffusion peut être anisotrope et hétérogène. Cette classe de méthodes localement conservatrices s’applique àune grande variété de mailles polyédriques non structurées qui peuvent ne pas se raccorder.La discrétisation en temps est totalement implicite. Nous obtenons un résultat de convergence basé sur des estimations a priori et sur l’application du théorème de compacité de Fréchet-Kolmogorov. Nous présentons aussi des tests numériques.Au Chapitre 3, nous discrétisons le problème de Signorini par un schéma de type gradient,qui s’écrit à l’aide d’une formulation variationnelle discrète et est basé sur des approximations indépendantes des fonctions et des gradients. On montre l’existence et l’unicité de la solution discrète ainsi que sa convergence vers la solution faible du problème continu. Nous présentons ensuite un schéma numérique basé sur la méthode SUSHI.Au Chapitre 4, nous appliquons un schéma semi-implicite en temps combiné avec la méthode SUSHI pour la résolution numérique d’un problème d’écoulements à densité variable ;il s’agit de résoudre des équations paraboliques de convection-diffusion pour la densité de soluté et le transport de la température ainsi que pour la pression. Nous simulons l’avance d’un front d’eau douce assez chaude et le transport de chaleur dans un aquifère captif qui est initialement chargé d’eau froide salée. Nous utilisons des maillages adaptatifs, basés sur des éléments de volume carrés. / This thesis bears on the modelling of groundwater flow and transport in porous media; we perform numerical simulations by means of finite volume methods and prove convergence results. In Chapter 1, we first apply a semi-implicit standard finite volume method and then the generalized finite volume method SUSHI for the numerical simulation of density driven flows in porous media; we solve a nonlinear convection-diffusion parabolic equation for the concentration coupled with an elliptic equation for the pressure. We apply the standard finite volume method to compute the solutions of a problem involving a rotating interface between salt and fresh water and of Henry's problem. We then apply the SUSHI scheme to the same problems as well as to a three dimensional saltpool problem. We use adaptive meshes, based upon square volume elements in space dimension two and cubic volume elements in space dimension three. In Chapter 2, we apply the generalized finite volume method SUSHI to the discretization of Richards equation, an elliptic-parabolic equation modeling groundwater flow, where the diffusion term can be anisotropic and heterogeneous. This class of locally conservative methods can be applied to a wide range of unstructured possibly non-matching polyhedral meshes in arbitrary space dimension. As is needed for Richards equation, the time discretization is fully implicit. We obtain a convergence result based upon a priori estimates and the application of the Fréchet-Kolmogorov compactness theorem. We implement the scheme and present numerical tests. In Chapter 3, we study a gradient scheme for the Signorini problem. Gradient schemes are nonconforming methods written in discrete variational formulation which are based on independent approximations of the functions and the gradients. We prove the existence and uniqueness of the discrete solution as well as its convergence to the weak solution of the Signorini problem. Finally we introduce a numerical scheme based upon the SUSHI discretization and present numerical results. In Chapter 4, we apply a semi-implicit scheme in time together with a generalized finite volume method for the numerical solution of density driven flows in porous media; it comes to solve nonlinear convection-diffusion parabolic equations for the solute and temperature transport as well as for the pressure. We compute the solutions for a specific problem which describes the advance of a warm fresh water front coupled to heat transfer in a confined aquifer which is initially charged with cold salt water. We use adaptive meshes, based upon square volume elements in space dimension two.