Título: High order well balanced numerical methods for a multilayer shallow-water model with variable density.
Doctorando: Ernesto Guerrero Fernández.
Directores: Manuel J. Castro Díaz y Tomás Morales de Luna. Departamento de Análisis Matemático, Estadística e Investigación Operativa, y Matemática Aplicada.
Centro: Universidad de Málaga.
Defensa: 3 de junio de 2022.
Calificación: Sobresaliente cum Laude. Mención Internacional.
This thesis presents some novel contributions within the framework of multilayer shallow-water models and well-balanced high order numerical schemes. In the first place, a multilayer shallow-water model with variable density is derived. A particular state equation is chosen such that the model can be written in terms of its relative density. The model is derived under the hydrostatic assumption for the pressure and consist on a system of partial differential equations with non-conservative products and source terms. As it is usual in multilayer systems, a closure relation is needed for the vertical transference terms, which are then expressed in terms of horizontal velocities. Finally, the model presents some stationary solutions that go beyond the usual lake-at-rest type stationary solutions. Since the model admits density fluctuations, stationary solutions corresponding to a vertical stratification in relative density are also possible. However, due to the multilayer shallow-water approach, the vertical profile depends on the given topography and the number of layers chosen. This may be seen as a discrete version for the multilayer approach for such vertical stratification profile.
A complete review of the existing techniques regarding finite volume methods based on path-conservative Riemann solvers and discontinuous Galerkin (DG) methods is also included. Of particular interest is the novel contribution of this thesis where a general strategy to design well-balanced DG methods for general systems of balanced laws is given. The strategy includes a limiting technique that preserves the well-balanced property of the numerical scheme. Moreover, it is compatible with several time discretization methods, such as ADER or Runge-Kutta. It has been successfully tested with the Burgers’ equation, the Euler equations with gravity and the one layer shallow-water equations.
The next step consists on providing numerical discretizations for the multilayer shallow-water model with variable density. The discretization proposed here is based on the finite volume method and the discontinuous Galerkin method, and both of them are novel contributions of this thesis. The finite volume method provides a second-order accurate approximation of the PDE system by means of a hydrostatic reconstruction that is able to preserve the water at rest solution with constant density and also, non-trivial stationary solutions corresponding to a vertical stratification of relative density. Likewise, the DG based numerical discretization allows for arbitrary high order in space and time in one step thanks to the use of the ADER-DG technique. This approach is also well-balanced for trivial and non-trivial stationary solutions. The limiting strategy is based on the MOOD paradigm, with a proper switching between the DG solver and a robust finite volume solver.
Finally, several numerical experiments are proposed that seek to prove the accuracy and robustness of the proposed numerical discretizations. They include a test studying the order of accuracy, where it can be seen that the desired order is achieved for both numerical solvers. Several well-balanced tests are also discussed. Non-trivial stationary solutions are not only preserved, but even recovered after a small perturbation. Next, several experiments are considered, including a lock exchange in relative density and a comparison with experimental data, where the numerical results yield excellent data agreement. The last experiment corresponds to a lock exchange in a two-dimensional channel.
As closure and in order to go beyond the results already presented, two appendices are included. The first one addresses the numerical implementation in two dimensional domains of the finite volume method proposed in this thesis for the multilayer shallow-water model with variable density. The second appendix explains the parallelization performed in CPU and in GPU, including a comparison of different parallelization strategies running in several hardware architectures.
Título: Reduced Basis Method applied to the Smagorinsky Turbulence Model.
Doctoranda: Cristina Caravaca García.
Directores: Tomás Chacón Rebollo y Macarena Gómez Mármol. Departamento de Ecuaciones Diferenciales y Análisis Numérico.
Centro: Universidad de Sevilla, Instituto de Matemáticas de la Universidad de Sevilla (IMUS).
Defensa: 3 de junio de 2022.
Calificación: Sobresaliente (provisional).
This PhD thesis addresses the numerical simulation of models that simulate the behavior of turbulent flows through reduced-order techniques. We use the Smagorinsky model, which is a large eddy simulation (LES) turbulence model. The primary objective of this dissertation is to build a posteriori error bounds for the finite element - reduced order solution of the Smagorinsky turbulence model in order to build the Reduced Basis (RB) Model.
The thesis is divided into two parts. In the first part, we seek for the application of a posteriori error bound estimator following the Brezzi-Rappaz-Raviart (BRR) theory of non-singular branches approximation of parametric non-linear problems to an optimal design problem of a cloister. In the second part, we seek to extend to the unsteady Smagorinsky model the techniques to construct RB solvers developed for steady problems and we propose another estimator based upon the Kolmogórov theory.
In Chapter 1, we introduce concepts that are used along the thesis such that difference ROM techniques, auxiliary results and the application of a posteriori error bound estimator for a 3D case. We consider the lid-driven cavity problem and we build a RB model, reducing the computational time from 2 hours to only 2 seconds.
Part I
Cloisters are a kind of courtyard widespread in old buildings such as cathedrals and monasteries. We are interested in the optimal thermal comfort design of the cloister, studying the dynamics of the air that flows into the courtyard by forced convection model, as a first approximation to our problem. We study how the dimensions of the corridor around the cloister affects to the flow in the cloister. To do this, we build a RB model using the width and the height of the corridor as parameters.
In Chapter 2, we consider the fluid dynamics only in 2D. We develop the RB problem building an a posteriori error bound estimator following the BRR theory for velocity and pressure. We are able to obtain the distribution of velocity, pressure, and temperature in barely 1,5 second in stead of 3,5 minutes with low error levels.
In Chapter 3, we add the temperature to the problem using a forced convection model. We build the RB problem developing an a posteriori error bound estimator for temperature, following now standard techniques since the problem is linear.
We design a functional to minimize the temperature in the ground floor in terms of the geometrical parameters and we computed the value of this functional for 625 pairs of parameters in less than 16 minutes. We obtain an optimum design that optimizes the thermal comfort in the lower part (to be occupied by people) of the cloister.
Part II
The main goal of this part is to build RB based upon error estimators for the unsteady Smagorinsky turbulence model.
In Chapter 4, we develop a priori estimates for velocity and pressure for the time-space discrete Smagorinsky model. These results are the key to apply the BRR theory in the next chapter.
In Chapter 5, we are able to apply the BRR theory for the development of a posteriori error bound estimator, not without finding difficulties. The key is the definition of the norm related to the problem. Unfortunately, the computation of the Stability Factor, necessary for the computation of the a posteriori estimator, is not low-cost, and it is necessary the implementation of further techniques to reduce the complexity.
In Chapter 6, we introduce the a posteriori error estimator based upon the Kolmogórov theory. The continuous problem should achieve a specific energy spectrum, since the full-order model is intended to be a good approximation of the continuous problem, it should also achieve this energy spectrum in the resolved part of the inertial spectrum. To validate the estimate, we develop an academic test in which we compare the use of the Kolmogórov estimator with the use of the error between the full order and reduced order solution to develop the RB problem. The test is based on a problem for which we obtain a velocity field that satisfies the −5∕3 law, and we build an RB problem using the POD+Greedy strategy. The number of basis functions and the error are similar to those obtained using the true error as “error estimator”, which indicates that the estimate works in this case.
All numerical tests presented in this thesis have been coded in FreeFem++ v. 4.8. Due to the complexity of the numerical tests, we use parallel computation using the PETSc package and MPI (Message Passing Interface). Every offline phase has been performed in the cluster Anonimus2021 which allows a High Performance Computation (HPC).