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1. Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

Author

Jonathan Jung, Stéphane Dellacherie, Pascal Omnes, Hydro-Québec - TransÉnergie et Équipement, DCMÉ, Prévisions de contrôle du réseau, Hydro-Québec TransÉnergie, Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Computational AGility for internal flows sImulations and compaRisons with Experiments (CAGIRE), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Pau et des Pays de l'Adour (UPPA), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, and PLAFRIM

Subjects
Numerical Analysis, Finite volume method, Applied Mathematics, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, Euler system, Numerical diffusion, Space (mathematics), 01 natural sciences, law.invention, Euler equations, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Mach number, law, Modeling and Simulation, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Cartesian coordinate system, 0101 mathematics, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

Published
2021
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2. Full discretization of time dependent convection-diffusion-reaction equation coupled with the Darcy system

Author

Nancy Chalhoub, Pascal Omnes, Toni Sayah, Rebecca El Zahlaniyeh, Université Saint-Joseph de Beyrouth (USJ), Service de Thermo-hydraulique et de Mécanique des Fluides (STMF), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Laboratoire Analyse, Géométrie et Applications (LAGA), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord

Subjects
a priori error estimates, Algebra and Number Theory, Discretization, Numerical analysis, finite element method, 010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, Backward Euler method, Darcy–Weisbach equation, Finite element method, convection-diffusion-reaction equation, 010101 applied mathematics, Computational Mathematics, Theory of computation, Applied mathematics, 0101 mathematics, Darcy's equations, Convection–diffusion equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; In this article, we study the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.

Published
2020
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3. Optimal absorption of acoustical waves by a boundary

Author

Pascal Omnes, Frédéric Magoulès, Thi Phuong Kieu Nguyen, Anna Rozanova-Pierrat, Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), CentraleSupélec, CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, CentraleSupélec-Université Paris-Saclay, Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, and Funding: This work was funded by the Pôle de Compétitivité Systematic (France) under thegrant OpenGPU, and the Pôle de Compétitivité CapDigital (France) under the grant Callisto-Sari.

Subjects
Absorption (acoustics), Control and Optimization, Helmholtz equation, Wave propagation, sound absorption, Boundary (topology), wave propagation, Robin boundary condition, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 35L405, 35J05, 35J25, 15A06, Shape optimization, fractals, 0101 mathematics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Dissipation, Lipschitz continuity, 010101 applied mathematics, Absorbing wall, shape optimization, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with d-set boundaries (N -- 1 $\le$ d < N). We introduce a class of admissible Lipschitz boundaries, in which an optimal shape of the wall exists in the following sense: We prove the existence of a Radon measure on this shape, greater than or equal to the usual Lebesgue measure, for which the corresponding solution of the Helmholtz problem realizes the infimum of the acoustic energy defined with the Lebesgue measure on the boundary. If this Radon measure coincides with the Lebesgue measure, the corresponding solution realizes the minimum of the energy. For a fixed porous material, considered as an acoustic absorbent, we derive the damping parameters of its boundary from the corresponding time-dependent problem described by the damped wave equation (damping in volume)., SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, In press

Published
2020
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4. Godunov type scheme for the linear wave equation with Coriolis source term

Author

Yohan Penel, Stéphane Dellacherie, Emmanuel Audusse, Pascal Omnes, and Do Minh Hieu

Subjects
Physics::Computational Physics, T57-57.97, Finite volume method, Applied mathematics. Quantitative methods, Discretization, Godunov's theorem, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, 01 natural sciences, Term (time), Mathematics::Numerical Analysis, 010101 applied mathematics, Linear map, symbols.namesake, Kernel (statistics), Froude number, symbols, QA1-939, 0101 mathematics, Mathematics
Abstract

We propose a method to explain the behaviour of the Godunov finite volume scheme applied to the linear wave equation with Coriolis source term at low Froude number. In particular, we use the Hodge decomposition and we study the properties of the modified equation associated to the Godunov scheme. Based on the structure of the discrete kernel of the linear operator discretized by using the Godunov scheme, we clearly explain the inaccuracy of the classical Godunov scheme at low Froude number and we introduce a way to modify it to recover a correct accuracy.

Published
2017

5. Numerical Results for a Discrete Duality Finite Volume Discretization Applied to the Navier–Stokes Equations

Author

Pascal Omnes, Sarah Delcourte, Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Service de Thermo-hydraulique et de Mécanique des Fluides (STMF), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Clément Cancès, and Pascal Omnes

Subjects
Finite volume method, Discretization, Mathematical analysis, Degrees of freedom (physics and chemistry), Duality (optimization), 010103 numerical & computational mathematics, Non-dimensionalization and scaling of the Navier–Stokes equations, 01 natural sciences, 010101 applied mathematics, symbols.namesake, Dirichlet boundary condition, Hagen–Poiseuille flow from the Navier–Stokes equations, symbols, 0101 mathematics, Navier–Stokes equations, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We present an application of the discrete duality finite volume method to the numerical approximation of the 2D Stokes or (unsteady) Navier–Stokes equations associated to Dirichlet boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as an extension of the classical MAC scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. Different numerical examples over triangular, cartesian, quadrangular and locally refined meshes are led in order to illustrate the possibilities and weaknesses of the method.

Published
2017
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6. Analysis of modified Godunov type schemes for the two-dimensional linear wave equation with Coriolis source term on cartesian meshes

Author

Yohan Penel, Minh Hieu Do, Emmanuel Audusse, Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Finite volume method, Physics and Astronomy (miscellaneous), Applied Mathematics, Courant–Friedrichs–Lewy condition, Godunov's scheme, 010103 numerical & computational mathematics, 01 natural sciences, Stability (probability), [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Computer Science Applications, law.invention, 010101 applied mathematics, Computational Mathematics, Nonlinear system, law, Modeling and Simulation, Applied mathematics, Cartesian coordinate system, 0101 mathematics, Shallow water equations, Geostrophic wind, Mathematics
Abstract

The study deals with collocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classic scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so, a Hodge-like decomposition is introduced. Then three different well-balanced strategies are introduced. Some properties of the associated modified equations are proven and then extended to the semi-discrete case. Stability of fully discrete schemes under a suitable CFL condition is established thanks to a Von Neumann analysis. Some numerical results reinforce the purpose and exhibit the concrete improvements achieved by the application of these new techniques in both linear and nonlinear cases.

Published
2017
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7. Analysis of Apparent Topography scheme for the linear wave equation with Coriolis force

Author

Yohan Penel, Emmanuel Audusse, Minh Hieu Do, Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Numerical Analysis, Geophysics and Ecology (ANGE), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Physics, Finite Volume Method, Coriolis Force, Finite volume method, Omega equation, Fluid mechanics, 010103 numerical & computational mathematics, Mechanics, Shallow water flows, 01 natural sciences, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 010305 fluids & plasmas, Physics::Geophysics, symbols.namesake, 0103 physical sciences, Fictitious force, symbols, 0101 mathematics, Kelvin wave, Shallow water equations, Pressure gradient, Geostrophic wind, Physics::Atmospheric and Oceanic Physics, Well-balanced Schemes
Abstract

International audience; The shallow water equations can be used to model many phenomena in geophysical fluid mechanics. For large scales, the Coriolis force plays an important role and the geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis force is an important feature. In this communication , we investigate the stability condition and the behavior of the so-called Apparent Topography scheme which is capable of capturing a discrete version of the geostrophic equilibrium.

Published
2017
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8. Finite Volumes for Complex Applications VIII - Hyperbolic, Elliptic and Parabolic Problems

Author

Pascal Omnes, Clément Cancès, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Service de Thermo-hydraulique et de Mécanique des Fluides (STMF), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, C. Cancès and P. Omnes, Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
010101 applied mathematics, Physics, Alternating direction implicit method, Elliptic partial differential equation, Discontinuous Galerkin method, Mathematical analysis, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010103 numerical & computational mathematics, Finite volume method for one-dimensional steady state diffusion, 0101 mathematics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience

Published
2017

9. Construction of modified Godunov type schemes accurate at any Mach number for the compressible Euler system

Author

Stéphane Dellacherie, Pascal Omnes, Pierre-Arnaud Raviart, Jonathan Jung, CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Computational Approximation with discontinous Galerkin methods and compaRison with Experiments (CAGIRE), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
linear wave equation, Applied Mathematics, Godunov's theorem, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, Euler system, Roe scheme, 01 natural sciences, Godunov scheme, 010101 applied mathematics, Roe solver, symbols.namesake, low Mach number flow, Mach number, Modeling and Simulation, Euler's formula, symbols, Compressibility, Compressible Euler system, 0101 mathematics, 10. No inequality, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Linear stability
Abstract

International audience; This article is composed of three self-consistent chapters that can be read independently of each other. In Chapter 1, we define and we analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Chapter 2, we apply the all Mach correction proposed in Chapter 1 to the case of the non-linear barotropic Euler system when the Godunov type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this non-linear case by showing how this construction is related with the linear analysis of Chapter 1. At last, we apply in Chapter 3 the all Mach correction proposed in Chapter 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Chapters 1, 2 and 3 justify the theoretical results and show that the obtained all Mach Godunov type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

Published
2016
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10. Preliminary results for the study of the Godunov Scheme Applied to the Linear Wave Equation with Porosity at Low Mach Number

Author

Pascal Omnes, Stéphane Dellacherie, Jonathan Jung, CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Laboratoire de recherche conventionné Modélisation et approximation numérique orientées pour l'énergie nucléaire (LRC Manon), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Computational Approximation with discontinous Galerkin methods and compaRison with Experiments (CAGIRE), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), AllianSTIC-EFREI, Efrei (Efrei), Département de Modélisation des Systèmes et Structures (DM2S), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
010103 numerical & computational mathematics, Space (mathematics), 01 natural sciences, 010305 fluids & plasmas, law.invention, Mathematics::Numerical Analysis, Godunov scheme, symbols.namesake, law, 0103 physical sciences, QA1-939, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Polygon mesh, Cartesian coordinate system, compressible Euler system with porosity, 0101 mathematics, Porosity, Linear wave equation, Mathematics, Physics::Computational Physics, T57-57.97, Applied mathematics. Quantitative methods, Godunov's theorem, Mathematical analysis, linear wave equation with porosity, Godunov's scheme, low Mach number flow, Mach number, all Mach scheme, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We introduce continuous tools to study the low Mach number behavior of the Godunov scheme applied to the linear wave equation with porosity on cartesian meshes. More precisely, we extend the Hodge decomposition to a weighted L2 space in the continuous case and we study the properties of the modified equation associated to this Godunov scheme. This allows to partly explain the inaccuracy of the Godunov scheme at low Mach number on cartesian meshes and to propose two corrections: a first one named low Mach and a second one named all Mach. These results are preliminary since it remains to prove them in the discrete case.

Published
2016
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11. A Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Laplace Equation

Author

Yohan Penel, Pascal Omnes, Yann Rosenbaum, Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13)

Subjects
MathematicsofComputing_NUMERICALANALYSIS, Duality (optimization), 010103 numerical & computational mathematics, 01 natural sciences, Mathematics::Numerical Analysis, Singular solution, AMS 65N15, 65N30, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Calculus, Applied mathematics, Polygon mesh, 0101 mathematics, ComputingMethodologies_COMPUTERGRAPHICS, Mathematics, Laplace's equation, Numerical Analysis, Finite volume method, Applied Mathematics, Numerical analysis, Mixed finite element method, discrete duality, 16. Peace & justice, a posteriori error estimation, Finite element method, 010101 applied mathematics, Computational Mathematics, finite volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], nonconforming meshes
Abstract

International audience; An efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the Laplace equation on very general twodimensional meshes. The main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework. Numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes.

Published
2009
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12. On the second-order convergence of a function reconstructed from finite volume approximations of the Laplace equation on Delaunay-Voronoi meshes

Author

Pascal Omnes, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), Service Fluide numériques, Modélisation et Etudes (SFME), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Delaunay meshes, finite volume method, 010103 numerical & computational mathematics, Topology, 01 natural sciences, symbols.namesake, Applied mathematics, Voronoi meshes, 0101 mathematics, Mathematics, Laplace's equation, Numerical Analysis, Partial differential equation, convergence, Laplace expansion, Applied Mathematics, Inverse Laplace transform, Laplace equation, Green's function for the three-variable Laplace equation, 010101 applied mathematics, Computational Mathematics, error estimates, Modeling and Simulation, Laplace transform applied to differential equations, Dirichlet boundary condition, symbols, Voronoi diagram, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken $P^1$ function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the~$L^2$ norm, under the sufficient condition that the right-hand side of the Laplace equation belongs to~$H^1(\Omega)$.

Published
2011
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13. A discrete duality finite volume approach to Hodge decomposition and div-curl problems on almost arbitrary two-dimensional meshes

Author

Komla Domelevo, Pascal Omnes, Sarah Delcourte, Laboratoire de Modélisation Physique et de l'Enrichissement (LMPE), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Institut Galilée-Université Paris 13 (UP13), CEA-Direction de l'Energie Nucléaire (CEA-DEN), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), and Service Fluide numériques, Modélisation et Etudes (SFME)

Subjects
Discretization, volumes finis, [PHYS.NUCL]Physics [physics]/Nuclear Theory [nucl-th], 65N06 (35F05), 010103 numerical & computational mathematics, [PHYS.NEXP]Physics [physics]/Nuclear Experiment [nucl-ex], 01 natural sciences, Mathematics::Numerical Analysis, Physics::Plasma Physics, Applied mathematics, Polygon mesh, 0101 mathematics, probleme div-rot, ComputingMilieux_MISCELLANEOUS, Mathematics, Curl (mathematics), Numerical Analysis, Finite volume method, convergence, Applied Mathematics, Numerical analysis, maillages non-conformes, Mathematical analysis, Differential operator, Physics::Classical Physics, maillages degeneres, 010101 applied mathematics, decomposition de Hodge discrete, Computational Mathematics, Vector field, Discrete differential geometry, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We define discrete differential operators such as grad, div and curl, on general two-dimensional non-orthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector fields, vector curls have a vanishing divergence and gradients have a vanishing curl. We apply these ideas to discretize div-curl systems. We give error estimates based on the reformulation of these systems into equivalent equations for the potentials. Numerical results illustrate the use of the method on several types of meshes, among which degenerating triangular meshes and non-conforming locally refined meshes.

Published
2007
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14. Equilibrated stress reconstructions for linear elasticity problems with application to a posteriori error analysis

Author

Alexandre Ern, Daniele Antonio Di Pietro, Rita Riedlbeck, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut des Sciences de la mécanique et Applications industrielles (IMSIA - UMR 9219), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC), Simulation for the Environment: Reliable and Efficient Numerical Algorithms (SERENA), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Clément Cancès, Pascal Omnes, and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)-Université Paris-Saclay-EDF R&D (EDF R&D)

Subjects
Arnold–Falk–Winther finite element, Cauchy stress tensor, Arnold–Winther finite element, Linear elasticity, linear elasticity, Geometry, 010103 numerical & computational mathematics, Mixed finite element method, 01 natural sciences, Symmetry (physics), Finite element method, Mathematics::Numerical Analysis, 010101 applied mathematics, Constraint (information theory), Stress (mechanics), Applied mathematics, A priori and a posteriori, equilibrated stress reconstruction, 0101 mathematics, a posteriori error estimate, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We present an a posteriori error estimate for the linear elasticity problem. The estimate is based on an equilibrated reconstruction of the Cauchy stress tensor, which is obtained from mixed finite element solutions of local Neumann problems. We propose two different reconstructions, one using Arnold–Winther mixed finite element spaces providing a symmetric stress tensor and one using Arnold–Falk– Winther mixed finite element spaces with a weak symmetry constraint. The performance of the estimate is illustrated on a numerical test with analytical solution.

Published
2017
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15. A splitting scheme for three-phase flow models

Author

Hamza Boukili, Jean-Marc Hérard, EDF (EDF), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), and Clément Cancès - Pascal Omnes

Subjects
Computer science, Numerical analysis, Three-phase flow, Three phase flow, 01 natural sciences, vapour explosion, 010305 fluids & plasmas, 010101 applied mathematics, Riemann hypothesis, symbols.namesake, Entropy inequality, 0103 physical sciences, symbols, Applied mathematics, shocks, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Data flow model, entropy, finite volumes, Step method
Abstract

International audience; A fractional step method that provides approximate solutions of a three-phase flow model is presented herein. The three-fluid model enables to handle smooth or discontinuous unsteady solutions. The numerical method is grounded on the use of the entropy inequality that governs smooth solutions of the set of PDEs. The evolution step relies on an explicit scheme, while implicit schemes are embedded in the relaxation step. The main properties of the scheme are given. Numerical approximations of two basic Riemann problems are eventually presented.

Published
2017
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16. A nonconforming high-order method for nonlinear poroelasticity

Author

Michele Botti, Daniele Antonio Di Pietro, Pierre Sochala, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Labex NUMEV (ANR-10-LABX-20) ref. 2014-2-006, HHOMM (ref. ANR-15-CE40-0005), Clément Cancès, Pascal Omnes, and ANR-15-CE40-0005,HHOMM,Méthodes hybrides d'ordre élevé sur maillages polyédriques(2015)

Subjects
Discretization, Poromechanics, Mathematical analysis, 010103 numerical & computational mathematics, Hybrid High-Order, Space (mathematics), 01 natural sciences, 65M08, 65N30, 74B20, 76S05, 010101 applied mathematics, Nonlinear system, Nonlinear poroelasticity, Operator (computer programming), Flow (mathematics), Discontinuous Galerkin method, General meshes, Polygon mesh, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, discontinuous Galerkin
Abstract

International audience; In this work, we introduce a novel algorithm for the quasi-static nonlin-ear poroelasticity problem describing Darcian flow in a deformable saturated porous medium. The nonlinear elasticity operator is discretized using a Hybrid High-Order method while the heterogeneous diffusion part relies on a Symmetric Weighted Interior Penalty discontinuous Galerkin scheme. The method is valid in two and three space dimensions, delivers an inf-sup stable discretization on general meshes including polyhedral elements and nonmatching interfaces, allows arbitrary approximation orders, and has a reduced cost thanks to the possibility of statically condensing a large subset of the unknowns for linearized versions of the problem. Moreover, the proposed construction can handle rough variations of the permeability coefficient and vanishing specific storage coefficient. Numerical tests demonstrating the performance of the method are provided.

Published
2017
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17. A Fractional Step Method to Simulate Mixed Flows in Pipes with a Compressible Two-Layer Model

Author

Jean-Marc Hérard, Stéphane Gerbi, Charles Demay, Christian Bourdarias, Benoît de Laage de Meux, EDF (EDF), Laboratoire de Mathématiques (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Clément Cancès - Pascal Omnes, and Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematical optimization, 010102 general mathematics, Two layer, 01 natural sciences, 010101 applied mathematics, mixed flow, Test case, Mixed flow, Two-layer model, Convergence (routing), Compressibility, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], implicit-explicit scheme, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Step method, Mathematics
Abstract

International audience; The so-called mixed flows in pipes include two-phase stratified regimes as well as single-phase pressurized regimes with transitions. It is proposed to handle those configurations numerically with the compressible two-layer model developed in [7]. Thus, a fractional step method is proposed to deal explicitly with the slow propagation phenomena and implicitly with the fast ones. It results in a large time-step scheme accurate in both regimes. Numerical experiments are performed including convergence results and academical test cases.

Published
2017
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18. Benchmark session: The 2D Hybrid High-Order method

Author

Daniele Antonio Di Pietro, Stella Krell, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Clément Cancès, Pascal Omnes, and ANR-15-CE40-0005,HHOMM,Méthodes hybrides d'ordre élevé sur maillages polyédriques(2015)

Subjects
Computer science, Mathematics::Analysis of PDEs, Degrees of freedom (statistics), 010103 numerical & computational mathematics, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Nonlinear system, Compressibility, Benchmark (computing), Order (group theory), Applied mathematics, Polygon mesh, Preprint, Session (computer science), 0101 mathematics, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We consider here the two-dimensional version of the Hybrid High-Order (HHO) method for the steady incompressible Navier–Stokes equations originally introduced in [Di Pietro, Krell, A Hybrid High-Order method for the steady incompressible Navier–Stokes problem, preprint arXiv:1607.08159 math.NA]. This method displays several advantageous features: it is inf-sup stable on general meshes including polyhedral elements and nonmatching interfaces, it supports arbitrary approximation order, and has a reduced computational cost thanks to the possibility of statically condensing a subset of both velocity and pressure degrees of freedom (DOFs) at each nonlinear iteration.

Published
2017
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19. Non-isothermal Compositional Two-Phase Darcy Flow: Formulation and Outflow Boundary Condition

Author

Farid Smaï, Laurence Beaude, Roland Masson, Simon Lopez, Konstantin Brenner, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Bureau de Recherches Géologiques et Minières (BRGM) (BRGM), Clément Cancès, Pascal Omnes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Laboratoire Jean Alexandre Dieudonné (LJAD), and ANR-16-CE06-0009,CHARMS,Modèles de Réservoirs Quantitatifs pour les Systèmes Hydrothermaux Complexes(2016)

Subjects
Convection, Darcy's law, Materials science, Boundary conditions for the interaction ground-atmosphere, business.industry, Geothermal energy, 010103 numerical & computational mathematics, Geophysics, Mechanics, 010502 geochemistry & geophysics, 01 natural sciences, 7. Clean energy, Isothermal process, Physics::Fluid Dynamics, Permeability (earth sciences), Finite volume scheme, Non-isothermal compositional two-phase Darcy flow model, Outflow, Boundary value problem, 0101 mathematics, Porous medium, business, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 0105 earth and related environmental sciences
Abstract

International audience; This article deals with the modelling and formulation of compositional gasliquid Darcy flow. Our model includes an advanced boundary condition at the interface between the porous medium and the atmosphere accounting for convective mass and energy transfer, liquid evaporation, and liquid outflow. The formulation is based on a fixed set of unknowns whatever the set of present phases. The thermodynamical equilibrium is expressed as complementary constraints. The model and its formulation are applied to the simulation of the Bouillante high energy geothermal field in Guadeloupe characterized by a high temperature closed to the surface.

Published
2017
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20. A Weighted Splitting Approach for Low-Mach Number Flows

Author

David Iampietro, Pascal Galon, Jean-Marc Hérard, Frédéric Daude, EDF (EDF), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Clément Cancès - Pascal Omnes, and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Physics, Convection, Shock wave, 010103 numerical & computational mathematics, State (functional analysis), Mechanics, 7. Clean energy, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Operator splitting, symbols.namesake, Mach number, Flow (mathematics), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics
Abstract

International audience; In steady-state regimes, water circulating in the nuclear power plants pipes behaves as a low Mach number flow. However, when steep phenomena occur, strong shock waves are produced. Herein, a fractional step approach allowing to decouple the convective from the acoustic effects is proposed. The originality is that the splitting between these two parts of the physics evolves dynamically in time according to the Mach number. The first one-dimensional explicit and implicit numerical results on a wide panel of Mach numbers show that this approach is as accurate and CPU-consuming as a state of the art Lagrange-Projection-type method.

Published
2017
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21. Uniform-in-Time Convergence of Numerical Schemes for a Two-Phase Discrete Fracture Model

Author

Julian Hennicker, Jérôme Droniou, Roland Masson, School of Mathematical Sciences [Clayton], Monash University [Clayton], TOTAL CSTJF, F-64018 Pau, France, COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (JAD), Pascal Omnes, Centre scientifique et Technique Jean Feger (CSTJF), TOTAL FINA ELF, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), and Laboratoire Jean Alexandre Dieudonné (LJAD)

Subjects
Discrete fracture model, Capillary pressure, Hydrogeology, Darcy's law, Weak solution, 010103 numerical & computational mathematics, Mechanics, Two-phase Darcy flow, 01 natural sciences, Uniform-in-time convergence, Gradient discretization method, Physics::Geophysics, 010101 applied mathematics, Matrix (mathematics), Flow (mathematics), Fracture (geology), 0101 mathematics, Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Geology
Abstract

International audience; Flow and transport in fracturedporous media are of paramount importance for many applications such as petroleum exploration and production, geological storage of carbon dioxide, hydrogeology, or geothermal energy. We consider here the two-phase discrete fracture model introduced in [3] which represents explicitly the fractures as codimension one surfaces immersed in the surrounding matrix domain. Then, the two-phase Darcy flow in the matrix is coupled with the two-phase Darcy flow in the fractures using transmission conditions accounting for fractures acting either as drains or barriers. The model takes into account complex networks of fractures, discontinuous capillary pressure curves at the matrix fracture interfaces and can be easily extended to account for gravity including in the width of the fractures. It also includes a layer of damaged rock at the matrix fracture interface with its own mobility and capillary pressure functions. In this work, the convergence analysis carried out in [3] in the framework of gradient discretizations [2] is extended to obtain the uniform-in-time convergence of the discrete solutions to a weak solution of the model.

Published
2017
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22. A Finite-Volume discretization of viscoelastic Saint-Venant equations for FENE-P fluids

Author

Sébastien Boyaval, Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), École des Ponts ParisTech (ENPC)-Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema)-EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), MATHematics for MatERIALS (MATHERIALS), Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Clément Cancès, Pascal Omnes, ANR-15-CE01-0013,SEDIFLO,Modélisation et simulation du transport solide en rivières(2015), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (Saint-Venant), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique (CERMICS), and École des Ponts ParisTech (ENPC)-École des Ponts ParisTech (ENPC)

Subjects
Suliciu relaxation scheme, Discretization, 010103 numerical & computational mathematics, 01 natural sciences, Viscoelasticity, 65N08, 010305 fluids & plasmas, FENE-P viscoelastic fluids, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Physics::Fluid Dynamics, Rheology, 0103 physical sciences, 0101 mathematics, Shallow water equations, Mathematics, Finite volume method, Mathematical analysis, Relaxation (iterative method), Physics::Classical Physics, MSC (2010): 65M08, Stability conditions, Nonlinear system, 35Q30, Saint-Venant equations, simple Riemann solver, Finite-Volume, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Saint-Venant equations can be generalized to account for a viscoelastic rheology in shallow flows. A Finite-Volume discretization for the 1D Saint-Venant system generalized to Upper-Convected Maxwell (UCM) fluids was proposed in Bouchut and Boyaval (M3AS 23(08): 1479–1526, 2013, [6]), which preserved a physically-natural stability property (i.e. free-energy dissipation) of the full system. It invoked a relaxation scheme of Suliciu type for the numerical computation of approximate solutions to Riemann problems. Here, the approach is extended to the 1D Saint-Venant system generalized to the finitely-extensible nonlinear elastic fluids of Peterlin (FENE-P). We are currently not able to ensure all stability conditions a priori, but the scheme is fully computable. And, using numerical simulations, it may help understand the famous High-Weissenberg number problem (HWNP) well-known in computational rheology.

Published
2017
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