In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

The Boussinesq approximation is a set of fluids equations utilized in the atmospheric and oceanographic sciences. They may be thought of as inhomogeneous, incompressible Euler or Navier-Stokes equations, where the inhomogeneous term is a scalar quantity, typically representing density or temperature, governed by a convection-diffusion equation. In this thesis, we prove local-in-time existence and uniqueness of an inviscid Boussinesq system. Furthermore, we show that under stronger assumptions, the local-in-time results can be extended to global-in-time existence and uniqueness as well. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov-type. We use paradifferential calculus and properties of the Besov-type spaces to control the growth of vorticity via an a priori estimate on the growth of density. This result is motivated by work of M. Vishik demonstrating local-in-time existence and uniqueness for 2D Euler equations in borderline Besov-type spaces, and by work of R. Danchin and M. Paicu showing the global well-posedness of the 2D Boussinesq system with initial data in critical Besov and Lp-spaces. / text

In this thesis, we analyze theoretical properties of the discontinuous Galerkin method (DGM) and propose novel approaches to implementation with the aim to increase its efficiency. First, we derive explicit expressions for the eigenvalues (spectrum) of the discontinuous Galerkin spatial discretization applied to the linear advection equation. We show that the eigenvalues are related to the subdiagonal [p/p+1] Pade approximation of exp(-z) when the p-th degree basis functions are used. Then, we extend the analysis to nonuniform meshes where both the size of elements and the composition of the mesh influence the spectrum. We show that the spectrum depends on the ratio of the size of the largest to the smallest cell as well as the number of cells of different types. We find that the spectrum grows linearly as a function of the proportion of small cells present in the mesh when the size of small cells is greater than some critical value. When the smallest cells are smaller than this critical value, the corresponding eigenvalues lie outside of the main spectral curve. Numerical examples on nonuniform meshes are presented to show the improvement on the time step restriction. In particular, this result can be used to improve the time step restriction on Cartesian grids. Finally, we present a discontinuous Galerkin method for solutions of the Euler equations on Cartesian grids with embedded geometries. Cutting an embedded geometry out of the Cartesian grid creates cut cells, which are difficult to deal with for two reasons. One is the restrictive CFL number and the other is the integration on irregularly shaped cells. We use explicit time integration employing cell merging to avoid restrictively small time steps. We provide an algorithm for splitting complex cells into triangles and use standard quadrature rules on these for numerical integration. To avoid the loss of accuracy due to straight sided grids, we employ the curvature boundary conditions. We show that the proposed method is robust and high-order accurate.

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Dans cette thèse, on étudie la contrôlabilité et la stabilisation de certaines équations aux dérivées partielles . On s'intéresse d'abord au problème du contrôle de l'équation d'Euler 3D incompressible par une force extérieure de dimension finie. Nous montrons que pour un choix approprié de l'espace de contrôle, la vitesse et la pression du fluide sont exactement contrôlables en projections. De plus, la vitesse est approximativement contrôlable. Nous montrons aussi que le système en question n'est pas exactement contrôlable par une force extérieure de dimension finie.On étudie aussi la contrôlabilité de l'équation d'Euler 3D compressible. Le contrôle est une force extérieure de dimension finie agissant uniquement sur l'équation de la vitesse. Nous montrons que la vitesse et la densité du fluide sont simultanément contrôlables. En particulier, le système est approximativement contrôlable et exactement contrôlable en projections. Dans la dernière partie, on étudie la stabilisation de l'équation d'Euler dans un cylindre infini.Nous montrons que pour toute solution stationnaire (c,0) du système d'Euler il existe un contrôle supporté dans une partie de la frontière du cylindre qui stabilise le système à (c,0). / In this thesis, we study the controllability and stabilization of certain partial differential equations.We consider first the problem of control of the 3D incompressible Euler equationby an external force of finite dimension. We show that for an appropriate choice of control space, the velocity and the pressure of the fluid are exactly controllable in projections.Moreover, the velocity is approximately controllable. We also show that the system in question is not exactly controllable by a finite-dimensional external force.We also study the controllability of the 3D compressible Euler equation. The control is a finite-dimensional external force acting only on the velocity equation. We show that the velocity and density of the fluid are simultaneously controllable. In particular, the system is approximately controllable and exactly controllable in projections.The last section of the thesis is devoted to the study of a stabilization problem for the 2D incompressible Euler system in an infinite strip with boundary controls. We show that for any stationary solution (c,0) of the Euler system there is a control which is supported in a given bounded part of the boundary of the strip and stabilizes the system to (c,0).

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No description available.

Riemann problems are of prime importance in computational fluid dynamics simulations using finite elements or finite volumes discretizations. In some applications, billions of Riemann problems might need to be solved in a single simulation, therefore it is important to have reliable and computationally efficient algorithms to do so. Given the nonlinearity of the flux function in most systems considered in practice, to obtain an exact solution for the Riemann problem explicitly is often not possible, and iterative solvers are required. However, because of issues found with existing iterative solvers like lack of convergence and high computational cost, their use is avoided and approximate solvers are preferred. In this thesis work, motivated by the advances in computer hardware and algorithms in the last years, we revisit the possibility of using iterative solvers to compute the exact solution for Riemann problems. In particular, we focus on the development, implementation, and performance comparison of iterative Riemann solvers for the shallow water and Euler equations. In a one-dimensional homogeneous framework for these systems, we consider several initial guesses and iterative methods for the computation of the Riemann solution. We find that efficient and reliable iterative solvers can be obtained by using recent estimates on the Riemann solution to modify and combine well-known methods. Finally, we consider the application of these solvers in finite volume simulations using the wave propagation algorithms implemented in Clawpack.

We develop and analyse explicit in time schemes for the computation of compressible flows, based on staggered in space. Upwinding is performed equation by equation only with respect to the velocity. The pressure gradient is built as the transpose of the natural divergence. For the barotropic Euler equations, the velocity convection is built to obtain a discrete kinetic energy balance, with residual terms which are non-negative under a CFL condition. We then show that, in 1D, if a sequence of discrete solutions converges to some limit, then this limit is the weak entropy solution. For the full Euler equations, we choose to solve the internal energy balance since a discretization of the total energy is rather unnatural on staggered meshes. Under CFL-like conditions, the density and internal energy are kept positive, and the total energy cannot grow. To obtain correct weak solutions with shocks satisfying the Rankine-Hugoniot conditions, we establish a kinetic energy identity at the discrete level, then choose the source term of the internal energy equation to recover the total energy balance at the limit. More precisely speaking, we prove that in 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit in the discrete kinetic and internal energy equations, we show that the limit of the sequence of solutions is a weak solution. Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections. In all chapters, we show numerical tests to illustrate for theoretical results. / We develop and analyse explicit in time schemes for the computation of compressible flows, based on staggered in space. Upwinding is performed equation by equation only with respect to the velocity. The pressure gradient is built as the transpose of the natural divergence. For the barotropic Euler equations, the velocity convection is built to obtain a discrete kinetic energy balance, with residual terms which are non-negative under a CFL condition. We then show that, in 1D, if a sequence of discrete solutions converges to some limit, then this limit is the weak entropy solution. For the full Euler equations, we choose to solve the internal energy balance since a discretization of the total energy is rather unnatural on staggered meshes. Under CFL-like conditions, the density and internal energy are kept positive, and the total energy cannot grow. To obtain correct weak solutions with shocks satisfying the Rankine-Hugoniot conditions, we establish a kinetic energy identity at the discrete level, then choose the source term of the internal energy equation to recover the total energy balance at the limit. More precisely speaking, we prove that in 1D, if we assume the L∞ and BV-stability and the convergence of the scheme, passing to the limit in the discrete kinetic and internal energy equations, we show that the limit of the sequence of solutions is a weak solution. Finally, we consider the computation of radial flows, governed by Euler equations in axisymetrical (2D) or spherical (3D) coordinates, and obtain similar results to the previous sections. In all chapters, we show numerical tests to illustrate for theoretical results.

Simulating transient compressible flows involving shock waves presents challenges to the CFD practitioner in terms of the mesh quality required to resolve discontinuities and prevent smearing. This document discusses a novel two-dimensional Cartesian anisotropic mesh adaptation technique implemented for transient compressible flow. This technique, originally developed for laminar incompressible flow, is efficient because it refines and coarsens cells using criteria that consider the solution in each of the cardinal directions separately. In this document the method will be applied to compressible flow. The procedure shows promise in its ability to deliver good quality solutions while achieving computational savings. Transient shock wave diffraction over a backward step and shock reflection over a forward step are considered as test cases because they demonstrate that the quality of the solution can be maintained as the mesh is refined and coarsened in time. The data structure is explained in relation to the computational mesh, and the object-oriented design and implementation of the code is presented. Refinement and coarsening algorithms are outlined. Computational savings over uniform and isotropic mesh approaches are shown to be significant.