Analysis of discrete finite element shallow-water models

Les équations de Saint-Venant sont un système aux dérivées partielles jouant un rôle central dans la modélisation des écoulements océaniques. La méthode des éléments finis est particulièrement adaptée pour résoudre les équations de Saint-Venant car elle offre une grande flexibilité sur les domaines irréguliers ainsi qu’une variété d’espaces pour l’approximation de la solution. Or, la qualité de la solution numérique dépend de l’interaction entre ces espaces. Pour certaines combinaisons ou paires d’élément finis la solution numérique peut présenter des oscillations articiellement introduites par la discrétisation. Cette thèse porte sur le comportement numérique des solutions aux équations de Saint-Venant obtenues par différentes paires d’éléments finis. Tout d’abord, une étude sur la dispersion des ondes d’inertie-gravité est présentée pour une sélection de neuf paires d’éléments finis. Un ensemble de trois propriétés est ensuite mis en évidence afin que la discrétisation respecte le comportement des équations analytiques. Une méthode basée sur le calcul des noyaux est utilisée pour caractériser les modes stationnaires correspondant aux écoulements géostrophiques. Finalement, les espaces vectoriels de Raviart-Thomas et Brezzi-Douglas-Marini sont analysés.The shallow-water equations system plays a central role in numerical oceanic models. The finite element method is particularly well suited to solve the shallow-water equations as it works on irregular meshes with a variety of approximation spaces. However, the behavior of the numerical solution highly depends on the interaction between these approximation spaces. For specific finite element pairs the solution may exhibit spurious oscillations induced by the discretization scheme. In this thesis, we analyze these oscillations for a wide selection of finite element pairs. The numerical dispersion of inertia-gravity waves is quantified with dispersion analyses. A constructive linear algebra approach is developed to compute the kernels of the discretized operators. The results are used to characterize the smallest representable vortices on both structured and unstructured meshes. A special attention is given to the Raviart-Thomas and Brezzi-Douglas-Marini approximation spaces

An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs

We extend the entropy stable high order nodal discontinuous Galerkin spectral element approximation for the non-linear two dimensional shallow water equations presented by Wintermeyer et al. [N. Wintermeyer, A. R. Winters, G. J. Gassner, and D. A. Kopriva. An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340:200-242, 2017] with a shock capturing technique and a positivity preservation capability to handle dry areas. The scheme preserves the entropy inequality, is well-balanced and works on unstructured, possibly curved, quadrilateral meshes. For the shock capturing, we introduce an artificial viscosity to the equations and prove that the numerical scheme remains entropy stable. We add a positivity preserving limiter to guarantee non-negative water heights as long as the mean water height is non-negative. We prove that non-negative mean water heights are guaranteed under a certain additional time step restriction for the entropy stable numerical interface flux. We implement the method on GPU architectures using the abstract language OCCA, a unified approach to multi-threading languages. We show that the entropy stable scheme is well suited to GPUs as the necessary extra calculations do not negatively impact the runtime up to reasonably high polynomial degrees (around $N=7$). We provide numerical examples that challenge the shock capturing and positivity properties of our scheme to verify our theoretical findings

Study on SPH Viscosity Term Formulations

For viscosity-dominated flows, the viscous effect plays a much more important role. Since the viscosity term in SPH-governing (Smoothed Particle Hydrodynamics) equations involves the discretization of a second-order derivative, its treatment could be much more challenging than that of a first-order derivative, such as the pressure gradient. The present paper summarizes a series of improved methods for modeling the second-order viscosity force term. By using a benchmark patch test, the numerical accuracy and efficiency of different approaches are evaluated under both uniform and non-uniform particle configurations. Then these viscosity force models are used to compute a documented lid-driven cavity flow and its interaction with a cylinder, from which the most recommended viscosity term formulation has been identified

Domain-specific implementation of high-order Discontinuous Galerkin methods in spherical geometry

In recent years, domain-specific languages (DSLs) have achieved significant success in large-scale efforts to reimplement existing meteorological models in a performance portable manner. The dynamical cores of these models are based on finite difference and finite volume schemes, and existing DSLs are generally limited to supporting only these numerical methods. In the meantime, there have been numerous attempts to use high-order Discontinuous Galerkin (DG) methods for atmospheric dynamics, which are currently largely unsupported in main-stream DSLs. In order to link these developments, we present two domain-specific languages which extend the existing GridTools (GT) ecosystem to high-order DG discretization. The first is a C++-based DSL called G4GT, which, despite being no longer supported, gave us the impetus to implement extensions to the subsequent Python-based production DSL called GT4Py to support the operations needed for DG solvers. As a proof of concept, the shallow water equations in spherical geometry are implemented in both DSLs, thus providing a blueprint for the application of domain-specific languages to the development of global atmospheric models. We believe this is the first GPU-capable DSL implementation of DG in spherical geometry. The results demonstrate that a DSL designed for finite difference/volume methods can be successfully extended to implement a DG solver, while preserving the performance-portability of the DSL.ISSN:0010-4655ISSN:1879-294

Mathematical and numerical modelling of dispersive water waves

Fecha de lectura de Tesis: 4 diciembre 2018.En esta tesis doctoral se expone en primer lugar una visión general del modelado de ondas dispersivas para la simulación de procesos tsunami-génicos. Se deduce un nuevo sistema bicapa con propiedades de dispersión mejoradas y un nuevo sistema hiperbólico. Además se estudian sus respectivas propiedades dispersivas, estructura espectral y ciertas soluciones analíticas. Así mismo, se ha diseñado un nuevo modelo de viscosidad sencillo para la simulación de los fenómenos físicos relacionados con la ruptura de olas en costa. Se establecen los resultados teóricos requeridos para el diseño de esquemas numéricos de tipo volúmenes finitos y Galerkin discontinuo de alto orden bien equilibrados para sistemas hiperbólicos no conservativos en una y dos dimensiones. Más adelante, los esquemas numéricos propuestos para los sistemas de presión no hidrostática introducidos se describen. Se pueden destacar diferentes enfoques y estrategias. Por un lado, se diseñan esquemas de volúmenes finitos implícitos de tipo proyección-corrección en mallas decaladas y no decaladas. Por otro lado, se propone un esquema numérico de tipo Galerkin discontinuo explícito para el nuevo sistema de EDPs hiperbólico propuesto. Para permitir simulaciones en tiempo real, una implementación eficiente en GPU de los métodos es llevado a cabo y algunas directrices sobre su implementación son dados. Los esquemas numéricos antes mencionados se han aplicado a test de referencia académicos y a situaciones físicas más desafiantes como la simulación de tsunamis reales, y la comparación con datos de campo. Finalmente, un último capítulo es dedicado a medir la influencia al considerar efectos dispersivos en la simulación de transporte y arrastre de sedimentos. Para ello, se deduce un nuevo sistema de dos capas de aguas someras, se diseña un esquema numérico y se muestran algunos test académicos y de validación, que ofrecen resultados prometedores

Weak Form of Stokes-Dirac Structures and Geometric Discretization of Port-Hamiltonian Systems

We present the mixed Galerkin discretization of distributed parameter port-Hamiltonian systems. On the prototypical example of hyperbolic systems of two conservation laws in arbitrary spatial dimension, we derive the main contributions: (i) A weak formulation of the underlying geometric (Stokes-Dirac) structure with a segmented boundary according to the causality of the boundary ports. (ii) The geometric approximation of the Stokes-Dirac structure by a finite-dimensional Dirac structure is realized using a mixed Galerkin approach and power-preserving linear maps, which define minimal discrete power variables. (iii) With a consistent approximation of the Hamiltonian, we obtain finite-dimensional port-Hamiltonian state space models. By the degrees of freedom in the power-preserving maps, the resulting family of structure-preserving schemes allows for trade-offs between centered approximations and upwinding. We illustrate the method on the example of Whitney finite elements on a 2D simplicial triangulation and compare the eigenvalue approximation in 1D with a related approach.Comment: Copyright 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0