Nouvelles méthodes numériques pour les écoulements en eaux peu profondes

This research project focuses on the development and evaluation of numerical methods for shallow flows by proposing new spatial and temporal discretization techniques. First, a new high-order explicit finite volume method and a class of semi-implicit schemes are introduced which are effective for modelling fast and slow waves in oceanic and atmospheric flows. In the second part of the research project, a central-upwind scheme is proposed for shallow water flows on variable topography using unstructured grids. In this part of the project, a new approach is proposed for the stability analysis of unstructured numerical schemes for shallow water equations. In the third part of the thesis, two finite volume methods are developed for the conservation laws on curved geometries which are potentially applicable to shallow flows on a sphere. For such cases, numerical schemes are developed by using the approach followed by Stanley Osher. This approach employs simple hyperbolic systems which generate complex wave phenomena, and solutions that are effective for assessing numerical methods. In our case, Burgers’ equations are used since they have played an important role in the development of shock-capturing schemes in fluid mechanics.Dans ce projet de recherche, on s'intéresse au développement et à l'évaluation de nouvelles méthodes numériques pour les écoulements peu profonds. De nouvelles techniques de discrétisation spatiales et temporelles des équations sont proposées. La première partie de la thèse est dédiée au développement d'une méthode des volumes finis explicite d'ordre élevé et d'une famille de schémas semi-implicites qui sont efficaces pour la modélisation des processus lents et rapides dans les écoulements océaniques et atmosphériques. La deuxième partie du projet de recherche concerne la construction d'un schéma numérique efficace sans solveur de Riemann pour les écoulements peu profonds avec une topographie variable sur un maillage non structuré. Dans cette partie de la thèse, une nouvelle approche est proposée pour l'analyse de stabilité des schémas numériques non structurés pour les équations en eaux peu profondes. Dans la troisième partie de la thèse, deux schémas de volumes finis sont développés pour les lois de conservation sur des surfaces courbes qui ont un large potentiel d'être appliqués aux écoulements peu profonds sur la sphère. Dans ces cas, les schémas numériques sont développés en adoptant la démarche suivie par Stanley Osher. Cette démarche consiste à utiliser des systèmes hyperboliques simples qui génèrent des phénomènes d'ondes complexes et des solutions qui ont différentes structures. Ces solutions sont très efficaces pour tester les méthodes numériques. Dans notre cas, nous avons utilisé les équations de Burgers qui ont joué un rôle très important dans le développement des schémas numériques à capture de chocs en mécanique des fluides

New numerical methods for shallow water flows

Dans ce projet de recherche, on s'intéresse au développement et à l'évaluation de nouvelles méthodes numériques pour les écoulements peu profonds. De nouvelles techniques de discrétisation spatiales et temporelles des équations sont proposées. La première partie de la thèse est dédiée au développement d'une méthode des volumes finis explicite d'ordre élevé et d'une famille de schémas semi-implicites qui sont efficaces pour la modélisation des processus lents et rapides dans les écoulements océaniques et atmosphériques. La deuxième partie du projet de recherche concerne la construction d'un schéma numérique efficace sans solveur de Riemann pour les écoulements peu profonds avec une topographie variable sur un maillage non structuré. Dans cette partie de la thèse, une nouvelle approche est proposée pour l'analyse de stabilité des schémas numériques non structurés pour les équations en eaux peu profondes. Dans la troisième partie de la thèse, deux schémas de volumes finis sont développés pour les lois de conservation sur des surfaces courbes qui ont un large potentiel d'être appliqués aux écoulements peu profonds sur la sphère. Dans ces cas, les schémas numériques sont développés en adoptant la démarche suivie par Stanley Osher. Cette démarche consiste à utiliser des systèmes hyperboliques simples qui génèrent des phénomènes d'ondes complexes et des solutions qui ont différentes structures. Ces solutions sont très efficaces pour tester les méthodes numériques. Dans notre cas, nous avons utilisé les équations de Burgers qui ont joué un rôle très important dans le développement des schémas numériques à capture de chocs en mécanique des fluides.This research project focuses on the development and evaluation of numerical methods for shallow flows by proposing new spatial and temporal discretization techniques. First, a new high-order explicit finite volume method and a class of semi-implicit schemes are introduced which are effective for modelling fast and slow waves in oceanic and atmospheric flows. In the second part of the research project, a central-upwind scheme is proposed for shallow water flows on variable topography using unstructured grids. In this part of the project, a new approach is proposed for the stability analysis of unstructured numerical schemes for shallow water equations. In the third part of the thesis, two finite volume methods are developed for the conservation laws on curved geometries which are potentially applicable to shallow flows on a sphere. For such cases, numerical schemes are developed by using the approach followed by Stanley Osher. This approach employs simple hyperbolic systems which generate complex wave phenomena, and solutions that are effective for assessing numerical methods. In our case, Burgers’ equations are used since they have played an important role in the development of shock-capturing schemes in fluid mechanics

Node-Diamond approximation of heterogeneous and anisotropic diffusion systems on arbitrary two-dimensional grids

We develop a new nodal numerical scheme for solving diffusion equations. Anisotropic and heterogeneous diffusion tensors are taken into account in these equations. The method allows to cover a wide range of general meshes such as non-confirming and distorted ones. The main idea consists in deriving the scheme from a discrete bilinear form using cellwise approximation of the diffusion tensor and particular discrete gradients. These gradients are conceived on diamonds partitioning the cell using local geometrical objects. The degrees of freedom are placed at the centers and vertices of cells. The cell unknowns can be eliminated without any fill-in. As a result, the coercivity of the scheme holds true unconditionally by construction. The convergence theorem of the Node-Diamond scheme is proved under classical assumptions on the physical parameters of the model equation and the mesh. Numerical results show the good behavior of the proposed approach on various examples among which we consider strongly anisotropic and heterogeneous systems. For instance, optimal accuracy consisting of quadratic rates for L2-errors and linear rates for H1-errors is obtained

Stability analysis of unstructured finite volume methods for linear shallow water flows using pseudospectra and singular value decomposition

The discretization of the shallow water system on unstructured grids can lead to spurious modes which usually can affect accuracy and/or cause stability problems. This paper introduces a new approach for stability analysis of unstructured linear finite volume schemes for linear shallow water equations with the Coriolis Effect using spectra, pseudospectra, and singular value decomposition. The discrete operator of the scheme is the principal parameter used in the analysis. It is shown that unstructured grids have a large influence on operator normality. In some cases the eigenvectors of the operator can be far from orthogonal, which leads to amplification of solutions and/or stability problems. Large amplifications of the solution can be observed, even for discrete operators which respect the condition of asymptotic stability, and in some cases even for Lax-Richtmyer stable methods. The pseudospectra are shown to be efficient for the verification of stability of finite volume methods for linear shallow water equations. In some cases, the singular value decomposition is employed for further analysis in order to provide more information about the existence of unstable modes. The results of the analysis can be helpful in choosing the type of mesh, the appropriate placements of the variables of the system on the grid, and the suitable discretization method which is stable for a wide range of modes.The authors thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper. This publication was made possible by NPRP Grant 4-935-2-354 from the Qatar National Research Fund (a member of the Qatar Foundation). The statements made herein are solely the responsibility of the authors. Appendix AScopu

Schemes with well-controlled dissipation. Hyperbolic systems in nonconservative form

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyper-bolic systems in nonconservative form —the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions

Sensitivity and Interdependency Analysis of the HBV Conceptual Model Parameters in a Semi-Arid Mountainous Watershed

Hydrological models, with different levels of complexity, have become inherent tools in water resource management. Conceptual models with low input data requirements are preferred for streamflow modeling, particularly in poorly gauged watersheds. However, the inadequacy of model structures in the hydrologic regime of a given watershed can lead to uncertain parameter estimation. Therefore, an understanding of the model parameters’ behavior with respect to the dominant hydrologic responses is of high necessity. In this study, we aim to investigate the parameterization of the HBV (Hydrologiska Byråns Vattenbalansavedelning) conceptual model and its influence on the model response in a semi-arid context. To this end, the capability of the model to simulate the daily streamflow was evaluated. Then, sensitivity and interdependency analyses were carried out to identify the most influential model parameters and emphasize how these parameters interact to fit the observed streamflow under contrasted hydroclimatic conditions. The results show that the HBV model can fairly reproduce the observed daily streamflow in the watershed of interest. However, the reliability of the model simulations varies from one year to another. The sensitivity analysis showed that each of the model parameters has a certain degree of influence on model behavior. The temperature correction factor (ETF) showed the lowest effect on the model response, while the sensitivity to the degree-day factor (DDF) highly depends on the availability of snow cover. Overall, the changes in hydroclimatic conditions were found to be mostly responsible for the annual variability of the optimal parameter values. Additionally, these changes seem to actuate the interdependency between the parameters of the soil moisture and the response routines, particularly Field Capacity (FC), the recession coefficient K0, the percolation coefficient (KPERC), and the upper reservoir threshold (UZL). The latter combines either to shrink the storage capacity of the model’s reservoirs under extremely high peak flows or to enlarge them under overestimated water supply, mainly provoked by abundant snow cover