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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. 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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3. 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

4. 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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5. 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

6. A discrete duality finite volume discretization of the vorticity-velocity-pressure stokes problem on almost arbitrary two-dimensional grids

Author

Sarah Delcourte and Pascal Omnes

Subjects
Numerical Analysis, Finite volume method, Applied Mathematics, Mathematical analysis, Degrees of freedom (physics and chemistry), Duality (optimization), Vorticity, Differential operator, Computational Mathematics, Convergence (routing), Partial derivative, Boundary value problem, Analysis, Mathematics
Abstract

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity-velocity-pressure formulation of the two-dimensional Stokes equations, associated to various nonstandard 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 Marker and Cell scheme to almost arbitrary meshes, thanks to an appropriate choice of degrees of freedom. The efficiency of the scheme is illustrated by numerical examples over unstructured triangular and locally refined nonconforming meshes, which confirm the theoretical convergence analysis led in the article. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1–30, 2015

Published
2014
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7. 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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8. 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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9. The influence of cell geometry on the Godunov scheme applied to the linear wave equation

Author

Felix Rieper, Pascal Omnes, and Stéphane Dellacherie

Subjects
Numerical Analysis, Physics and Astronomy (miscellaneous), Discretization, Applied Mathematics, Operator (physics), Godunov's theorem, Mathematical analysis, Godunov's scheme, Mathematics::Numerical Analysis, Computer Science Applications, Computational Mathematics, symbols.namesake, Kernel (image processing), Modeling and Simulation, Euler's formula, symbols, Compressibility, Tetrahedron, Mathematics
Abstract

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, the Godunov scheme applied to the resolution of the compressible Euler or Navier-Stokes system is accurate at low Mach number when the mesh is triangular or tetrahedral and is not accurate when the mesh is a 2D (or 3D) cartesian mesh. This approach confirms also the fact that a Godunov scheme remains accurate when it is modified by simply centering the discretization of the pressure gradient.

Published
2010
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10. A finite volume method for the approximation of Maxwell’s equations in two space dimensions on arbitrary meshes

Author

Pascal Omnes, S. Layouni, and F. Hermeline

Subjects
Numerical Analysis, Finite volume method, Physics and Astronomy (miscellaneous), Discretization, Applied Mathematics, Mathematical analysis, Gauss, Degrees of freedom (physics and chemistry), Duality (optimization), Differential operator, Topology, Computer Science Applications, Computational Mathematics, symbols.namesake, Maxwell's equations, Modeling and Simulation, symbols, Gauss's law, Mathematics
Abstract

A new finite volume method is presented for discretizing the two-dimensional Maxwell equations. This method may be seen as an extension of the covolume type methods to arbitrary, possibly non-conforming or even non-convex, n-sided polygonal meshes, thanks to an appropriate choice of degrees of freedom. An equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles. The main properties of the scheme are its energy conservation, its stability under a CFL-like condition, and the fact that it preserves Gauss' law and divergence free magnetic fields. Second-order convergence is demonstrated numerically on non-conforming and distorted meshes.

Published
2008
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12. An a Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Stokes Equations

Author

Anh Ha Le, 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, and Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, MSC 65N08, 65N15, 76D07, Finite volume method, Discretization, Adaptive mesh refinement, Applied Mathematics, Duality (mathematics), Mathematical analysis, Estimator, Stokes equations, discrete duality, Upper and lower bounds, a posteriori error estimation, stabilization, Computational Mathematics, Modeling and Simulation, Polygon mesh, Constant (mathematics), Analysis, finite volumes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; We derive an a posteriori error estimation for the discrete duality finite volume (DDFV) discretization of the stationary Stokes equations on very general twodimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the velocity and one for the error on the pressure. They both include a contribution related to the error due to the stabilization of the scheme, and a contribution due to the discretization itself. The estimators are globally upper as well as locally lower bounds for the errors of the DDFV discretization. They are fully computable as soon as a lower bound for the inf-sup constant is available. Numerical experiments illustrate the theoretical results and we especially consider the influence of the penalty parameter on the error for a fixed mesh and also of the mesh size for a fixed value of the penalty parameter. A global error reducing strategy that mixes the decrease of the penalty parameter and adaptive mesh refinement is described.

Published
2015
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13. Divergence Correction Techniques for Maxwell Solvers Based on a Hyperbolic Model

Author

U. Voβ, Claus-Dieter Munz, Eric Sonnendrücker, Rudolf Schneider, and Pascal Omnes

Subjects
Electromagnetic field, Numerical Analysis, Gauss's law for gravity, Physics and Astronomy (miscellaneous), Applied Mathematics, Mathematical analysis, Inhomogeneous electromagnetic wave equation, Computer Science Applications, Computational Mathematics, symbols.namesake, Continuity equation, Maxwell's equations, Modeling and Simulation, symbols, Gauss's law, Poisson's equation, Hyperbolic partial differential equation, Mathematics
Abstract

Usually, non-stationary numerical calculations in electromagnetics are based on the hyperbolic evolution equations for the electric and magnetic fields and leave Gauss' law out of consideration because the latter is a consequence of the former and of the charge conservation equation in the continuous case. However, in the simulation of the self-consistent movement of charged particles in electromagnetic fields, it is a well-known fact that the approximation of the particle motion introduces numerical errors and that, consequently, the charge conservation equation is not satisfied on the dicrete level. Then, in order to avoid the increase of errors in Gauss' law, a divergence cleaning step which solves a Poisson equation for a correction potential is often added. In the present paper, a new method for incorporating Gauss' law into non-stationary electromagnetic simulation codes is developed, starting from a constrained formulation of the Maxwell equations. The resulting system is hyperbolic, and the divergence errors propagate with the speed of light to the boundary of the computational domain. Furthermore, the basic ideas of the numerical approximation are introduced and the extended hyperbolic system is treated numerically within the framework of high-resolution finite-volume schemes. Simulation results obtained with this new technique for pure electromagnetic wave propagation and for an electromagnetic particle-in-cell computation are presented and compared with other methods.

Published
2000
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14. On the Godunov Scheme Applied to the Variable Cross-Section Linear Wave Equation

Author

Stéphane Dellacherie and Pascal Omnes

Subjects
symbols.namesake, Cross section (physics), Mach number, Scheme (mathematics), Godunov's theorem, Mathematical analysis, symbols, Godunov's scheme, Constant (mathematics), Wave equation, Mathematics, Variable (mathematics)
Abstract

We investigate the accuracy of the Godunov scheme applied to the variable cross-section acoustic equations. Contrarily to the constant cross-section case, the accuracy issue of this scheme in the low Mach number regime appears even in the one-dimensional case; on the other hand, we show that it is possible to construct another Godunov type scheme which is accurate in the low Mach number regime.

Published
2011
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15. 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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16. Enforcing Gauss’ Law in Computational Elec-Tromagnetics Within a Finite-Volume Framework

Author

Pascal Omnes, Rudolf Schneider, and Claus-Dieter Munz

Subjects
symbols.namesake, Gauss's law for gravity, Finite volume method, Maxwell's equations, Gauss, Scattering-matrix method, Mathematical analysis, symbols, Computational electromagnetics, Gauss's law, Spurious relationship, Mathematics
Abstract

The problem of spurious solutions due to the violation of Gauss’ law in computational electromagnetics is avoided by solving an equivalent Maxwell system that takes this constraint into account. A second-order accurate finite-volume method is proposed to solve this linear, first-order strictly hyperbolic reformulated system. Numerical examples demonstrate the validity of this approach.

Published
2001
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17. 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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18. 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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