A large time-step and well-balanced Lagrange-Projection type scheme for the shallow-water equations

This work focuses on the numerical approximation of the Shallow Water Equations (SWE) using a Lagrange-Projection type approach. We propose to extend to this context recent implicit-explicit schemes developed in the framework of compressibleflows, with or without stiff source terms. These methods enable the use of time steps that are no longer constrained by the sound velocity thanks to an implicit treatment of the acoustic waves, and maintain accuracy in the subsonic regime thanks to an explicit treatment of the material waves. In the present setting, a particular attention will be also given to the discretization of the non-conservative terms in SWE and more specifically to the well-known well-balanced property. We prove that the proposed numerical strategy enjoys important non linear stability properties and we illustrate its behaviour past several relevant test cases

Shape sensitivity analysis in aerodynamics using an isogeometric Discontinuous Galerkin method

International audienceThe sensitivity equation method aims at estimating the derivative of the solution of partial differential equations with respect to a parameter of interest. The objective of this work is to investigate the ability of an isogeometric Discontinuous Galerkin (DG) method to evaluate accurately sensitivities with respect to shape parameters originating from Computer-Aided Design (CAD), in the context of compressible aerodynamics. The isogeometric DG method relies on Non- Uniform Rational B-Spline representations, which allow to define a high-order numerical scheme for Euler/Navier-stokes equations, fully consistent with CAD geometries. We detail how this formulation can be exploited to construct an efficient and accurate approach to evaluate shape sensitivities. A particular attention is paid to the treatment of boundary conditions for sensitivities, which are more tedious in the case of geometrical parameters. The proposed methodology is first verified on a test- case with analytical solution and then applied to two more demanding problems, that concern the inviscid flow around an airfoil with its camber as shape parameter and the unsteady viscous flow around a three-element airfoil with the positions of slat and flap as parameters

A NURBS-based Discontinuous Galerkin method for CAD compliant flow simulations

International audienceIn this work, we explain how a classical nodal Discontinuous Galerkin (DG) method for conservation laws can be modified to be geometrically exact with respect to CAD (Computer-Aided Design) data. The proposed approach relies on the use of rational Bézier elements, that can exactly match geometries defined by NURBS (Non-Uniform Rational B-Splines) after some basic transformations. It has been found convenient to use the same basis to describe the solution, yielding a so-called isogeometric formulation. The resulting method exhibits optimal convergence rates and facilitates couplings with geometry, e.g. for local refinement, shape sensitivity analysis, or moving computational domains. Illustrations are provided for two-dimensional compressible Euler and Navier-Stokes equations

Simulation numérique d'écoulements compressibles complexes par des méthodes de type Lagrange-projection : applications aux équations de Saint-Venant

In this thesis we study a family of numerical schemes solving the shallow water equations system. These schemes use a Lagrange-projection like splitting operator technique in order to separate the gravity waves and the transport waves. An implicit-explicit treatment of the acoustic system (linked to the gravity waves) allows the schemes to stay stable with large time step. The correction of the pressure fluxes enables the obtain of a precise approximation solution whatever the regime flow is with respect to the Froude number. A particular attention has been paid over the source term treatment which permits to take the topography into account. We especially obtain the so-called well-balanced property giving the exact conservation of some steady states, namely the "lake at rest" state. 1D and 2D versions of this methods have been studied and implemented in the finite volumes framework. Finally, a high order discontinuous Galerkin extension has been proposed in 1D with classical limiters along with a combined MOOD loop a posteriori limiting strategy.On étudie dans le cadre de la thèse une famille de schémas numériques permettant de résoudre les équations de Saint-Venant. Ces schémas utilisent une décomposition d'opérateur de type Lagrange-projection afin de séparer les ondes de gravité et les ondes de transport. Un traitement implicite du système acoustique (relié aux ondes de gravité) permet aux schémas de rester stable avec de grands pas de temps. La correction des flux de pression rend possible l'obtention d'une solution approchée précise quel que soit le régime d'écoulement vis-à-vis du nombre de Froude. Une attention toute particulière est portée sur le traitement du terme source qui permet la prise en compte de l'influence de la topographie. On obtient notamment la propriété dite équilibre permettant de conserver exactement certains états stationnaires, appelés état du "lac au repos". Des versions 1D et 2D sur maillages non-structurés de ces méthodes ont été étudiées et implémentées dans un cadre volumes finis. Enfin, une extension vers des méthodes ordres élevés Galerkin discontinue a été proposée en 1D avec des limiteurs classiques ainsi que combinée avec une boucle MOOD de limitation a posteriori

Numerical simulation of complex compressible flows by Lagrange-projection type methods : applications to shallow water equations

On étudie dans le cadre de la thèse une famille de schémas numériques permettant de résoudre les équations de Saint-Venant. Ces schémas utilisent une décomposition d'opérateur de type Lagrange-projection afin de séparer les ondes de gravité et les ondes de transport. Un traitement implicite du système acoustique (relié aux ondes de gravité) permet aux schémas de rester stable avec de grands pas de temps. La correction des flux de pression rend possible l'obtention d'une solution approchée précise quel que soit le régime d'écoulement vis-à-vis du nombre de Froude. Une attention toute particulière est portée sur le traitement du terme source qui permet la prise en compte de l'influence de la topographie. On obtient notamment la propriété dite équilibre permettant de conserver exactement certains états stationnaires, appelés état du "lac au repos". Des versions 1D et 2D sur maillages non-structurés de ces méthodes ont été étudiées et implémentées dans un cadre volumes finis. Enfin, une extension vers des méthodes ordres élevés Galerkin discontinue a été proposée en 1D avec des limiteurs classiques ainsi que combinée avec une boucle MOOD de limitation a posteriori.In this thesis we study a family of numerical schemes solving the shallow water equations system. These schemes use a Lagrange-projection like splitting operator technique in order to separate the gravity waves and the transport waves. An implicit-explicit treatment of the acoustic system (linked to the gravity waves) allows the schemes to stay stable with large time step. The correction of the pressure fluxes enables the obtain of a precise approximation solution whatever the regime flow is with respect to the Froude number. A particular attention has been paid over the source term treatment which permits to take the topography into account. We especially obtain the so-called well-balanced property giving the exact conservation of some steady states, namely the "lake at rest" state. 1D and 2D versions of this methods have been studied and implemented in the finite volumes framework. Finally, a high order discontinuous Galerkin extension has been proposed in 1D with classical limiters along with a combined MOOD loop a posteriori limiting strategy

Shape Sensitivity Analysis for Hyperbolic Systems using an Isogeometric Discontinuous Galerkin Method

International audienc

Simulation numérique d'écoulements compressibles complexes par des méthodes de type Lagrange-projection : applications aux équations de Saint-Venant

In this thesis we study a family of numerical schemes solving the shallow water equations system. These schemes use a Lagrange-projection like splitting operator technique in order to separate the gravity waves and the transport waves. An implicit-explicit treatment of the acoustic system (linked to the gravity waves) allows the schemes to stay stable with large time step. The correction of the pressure fluxes enables the obtain of a precise approximation solution whatever the regime flow is with respect to the Froude number. A particular attention has been paid over the source term treatment which permits to take the topography into account. We especially obtain the so-called well-balanced property giving the exact conservation of some steady states, namely the "lake at rest" state. 1D and 2D versions of this methods have been studied and implemented in the finite volumes framework. Finally, a high order discontinuous Galerkin extension has been proposed in 1D with classical limiters along with a combined MOOD loop a posteriori limiting strategy.On étudie dans le cadre de la thèse une famille de schémas numériques permettant de résoudre les équations de Saint-Venant. Ces schémas utilisent une décomposition d'opérateur de type Lagrange-projection afin de séparer les ondes de gravité et les ondes de transport. Un traitement implicite du système acoustique (relié aux ondes de gravité) permet aux schémas de rester stable avec de grands pas de temps. La correction des flux de pression rend possible l'obtention d'une solution approchée précise quel que soit le régime d'écoulement vis-à-vis du nombre de Froude. Une attention toute particulière est portée sur le traitement du terme source qui permet la prise en compte de l'influence de la topographie. On obtient notamment la propriété dite équilibre permettant de conserver exactement certains états stationnaires, appelés état du "lac au repos". Des versions 1D et 2D sur maillages non-structurés de ces méthodes ont été étudiées et implémentées dans un cadre volumes finis. Enfin, une extension vers des méthodes ordres élevés Galerkin discontinue a été proposée en 1D avec des limiteurs classiques ainsi que combinée avec une boucle MOOD de limitation a posteriori

Shape Sensitivity Analysis for Hyperbolic Systems using an Isogeometric Discontinuous Galerkin Method

International audienc

A well-balanced Discontinuous-Galerkin Lagrange-Projection scheme for the Shallow Water Equations

This work considers the Shallow Water equations (SWE) and proposes a high order conservative scheme based on a Lagrange-Projection decomposition. The high order in space and time are achieved using Discontinuous-Galerkin (DG) and Runge-Kutta (RK) strategies. The use of a Lagrange-Projection decomposition enables the use of time steps that are not constrained by the sound speed thanks to an implicit treatment of the acoustic waves (Lagrange step), while the transport waves (Projection step) are treated explicitly. We prove that our scheme satisfies the well-balanced property as well as non linear stability properties. Numerical evidences are also given