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5. Analysis of compressible bubbly flows. Part I: Construction of a microscopic model

Author

Matthieu Hillairet, Hélène Mathis, and Nicolas Seguin

Abstract

In this note, we introduce a microscopic model for the motion of gas bubbles in a viscous fluid. By interpreting a bubble as a compressible fluid with infinite shear viscosity, we derive a pde/ode system coupling the density/velocity/pressure in the surrounding fluid with the linear/angular velocities and radii of the bubbles. We provide a 1D analogue of the system and construct an existence theory for this simplified system in a natural regularity framework. The second part of the paper is a preparatory work for the derivation of an averaged or macroscopic model.

Published
2023
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16. A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary

Author

Benjamin Boutin, Thi Hoai Thuong Nguyen, Nicolas Seguin, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-17-CE40-0025, Agence Nationale de la Recherche, Grant 642768 (ModCompShock), Innovative Training Networks, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), European Project: 642768,H2020,H2020-MSCA-ITN-2014,ModCompShock(2015), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
Boundary (topology), 010103 numerical & computational mathematics, summation by parts operators, Space (mathematics), 01 natural sciences, Stability (probability), 010305 fluids & plasmas, central schemes, 0103 physical sciences, medicine, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], damped wave equation, Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, Laplace transform, energy estimates, Applied Mathematics, Mathematical analysis, Stiffness, Damped wave, Computational Mathematics, Modeling and Simulation, hyperbolic relaxation system, Relaxation (approximation), medicine.symptom, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis
Abstract

International audience; We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of the initial boundary value problem for relaxation system independent of stiffness of the source term and of the space step. The boundary is approximated using a summation-by-parts method and the stiff stability is proved by energy estimates and Laplace transform. We also investigate if the condition is also necessary, following the continuous case studied by Xin and Xu (2000).; Nous étudions la stabilité du schéma semi-discret centré pour l'équation des ondes linéaire amortie posé sur un demi-espace. Nous dégageons une condition suffisante portant sur la condition de bord, pour la stabilité du problème semi-discret avec donnée initiale et donnée de bord, ceci de manière uniforme par rapport à la raideur du terme source de relaxation ainsi qu'au pas d'espace. La discrétisation de la condition de bord employée provient de l'approche SBP et l'uniforme stabilité s'obtient par l'utilisation de méthodes d'énergie et de la transformée de Laplace. Nous examinons également au travers d'expériences numériques le caractère nécessaire de la condition retenue, de sorte à confronter notre résultat à l'étude de Xin et Xu (2000) portant sur le cas continu.

Published
2020
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18. Stability of stationary solutions of singular systems of balance laws

Author

Nicolas Seguin, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Kullback–Leibler divergence, 010103 numerical & computational mathematics, 01 natural sciences, Hyperbolic systems, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, Mathematics (miscellaneous), Exponential stability, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, stationary state, Entropy (arrow of time), Mathematical Physics, Mathematics, Finite volume method, non-conservative systems, Applied Mathematics, Numerical analysis, relative entropy, Fluid mechanics, Numerical Analysis (math.NA), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, Dissipation, 010101 applied mathematics, well-balanced schemes, finite volume schemes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Stationary state, Analysis of PDEs (math.AP)
Abstract

International audience; The stability of stationary solutions of first-order systems of PDE's are considered. They may include some singular geometric terms, leading to discontinuous flux and non-conservative products. Based on several examples in Fluid Mechanics, we assume that these systems are endowed with a partially convex entropy. We first construct an associated relative entropy which allows to compare two states which share the same geometric data. This way, we are able to prove the stability of some stationary states within entropy weak solutions. This result applies for instance to the shallow-water equations with bathymetry. Besides, this relative entropy can be used to study finite volume schemes which are entropy-stable and well-balanced, and due to the numerical dissipation inherent to these methods, asymptotic stability of discrete stationary solutions is obtained. This analysis does not make us of any specific definition of the non-conservative products, applies to non-strictly hyperbolic systems, and is fully multidimensional with unstructured meshes for the numerical methods.

Published
2019
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20. Asymptotic preserving discretisation of a Jin–Xin model with implicit equilibrium manifold on a bounded domain

Author

Nicolas Seguin, Magali Tournus, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), École Centrale de Marseille (ECM), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)

Subjects
Hyperbolic Relaxation, Implicit function, Discretization, Applied Mathematics, General Mathematics, Linear system, Asymptotic Preserving scheme, 010103 numerical & computational mathematics, 01 natural sciences, Domain (mathematical analysis), Manifold, 010101 applied mathematics, Computational Mathematics, Boundary layer, Bounded function, Applied mathematics, Limit (mathematics), Boundary value problem, 0101 mathematics, [MATH]Mathematics [math], Mathematics
Abstract

In this paper, we design and analyze a numerical scheme that approximates a Jin–Xin linear system with implicit equilibrium on a bounded domain. This scheme relaxes toward the asymptotic limit of the linear system. The main properties of the limiting scheme are that it does not require to invert the implicit function defining the manifold and that it provides an accurate discretization of the boundary conditions.

Published
2020
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21. Some mathematical properties of a barotropic multiphase flow model

Author

Nicolas Seguin, Khaled Saleh, Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), 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 Claude Bernard Lyon 1 (UCBL), Université de Lyon, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-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)-Centre National de la Recherche Scientifique (CNRS), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), and Saleh, Khaled

Subjects
T57-57.97, Applied mathematics. Quantitative methods, Hyperbolic PDEs, Entropy, Multiphase flow, Mathematical properties, MSC: 76T10,35L60,35Q35,35F55, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Symmetrizable systems, Convexity, Physics::Fluid Dynamics, Multiphase flows, Barotropic fluid, Compressibility, QA1-939, Entropy (information theory), Compressible flows, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL] Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Statistical physics, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We study a model for compressible multiphase flows involving N non miscible barotopic phases where N is arbitrary. This model boils down to the barotropic Baer-Nunziato model when N = 2. We prove the weak hyperbolicity property, the non-strict convexity of the natural mathematical entropy, and the existence of a symmetric form.

Published
2020
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23. Dissipative formulation of initial boundary value problems for Friedrichs’ systems

Author

Bruno Després, Clément Mifsud, Nicolas Seguin, Sorbonne Université (SU), 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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), 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, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Constant coefficients, Picard–Lindelöf theorem, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mixed boundary condition, Space (mathematics), Wave equation, 01 natural sciences, Robin boundary condition, 010101 applied mathematics, Dissipative system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Analysis, Mathematics
Abstract

In this article we present a dissipative definition of a solution for initial boundary value problems for Friedrichs’ systems posed in the space . We study the information contained in this definition and prove an existence and uniqueness theorem in the non-characteristic case and with constant coefficients. Finally, we compare our choice of boundary condition to previous works, especially on the wave equation and show how to model additional constrained problems in view of initial boundary value problems for viscoplastic equations.

Published
2015
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24. On Well-Posedness for a Multi-particle Fluid Model

Author

Jens Klotzky, Nicolas Seguin, and Christian Klingenberg

Subjects
Shock wave, Physics, 010102 general mathematics, Lambda, Lipschitz continuity, 01 natural sciences, Burgers' equation, 010101 applied mathematics, Drag, Initial value problem, 0101 mathematics, Finite set, Well posedness, Mathematical physics
Abstract

In this paper, we study a one-dimensional fluid modelled by the Burgers equation influenced by an arbitrary but finite number of particles N(t) moving inside the fluid, each one acting as a point-wise drag force with a particle-related friction constant \(\lambda \). For given particle paths \(h_i(t)\), we only assume finite speed of particles, allowing for crossing, merging and splitting of particles. This model is an extension of existing models for fluid interactions with a single particle; compare (Andreianov et al., SIAM J Math Anal 46(2):1030–1052, 2014, [3], Lagoutiere et al., J Differ Equ 245(11):3503–3544, 2008, [10]): $$ \partial _t u(x,t) + \partial _x \left( \frac{u^2}{2}\right) = \sum _{i=1}^N \lambda (h_i'(t)-u(t,h_i(t))\delta (x-h_i(t))$$ Well-posedness for the Cauchy problem, as well as an \(L^\infty \) bound, is proven under the weak assumption that particle paths are Lipschitz continuous. In this context, an entropy admissibility criteria are shown, using the theory of \(L^1\)-dissipative germs, compare (Andreianov et al., Arch Ration Mech Anal 201:26–86, 2011, [2]), to deal with the moving interfaces resulting from the point-wise particles and the shock waves from the fluid equation interacting with them.

Published
2018
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25. Compressible Heterogeneous Two-Phase Flows

Author

Nicolas Seguin

Subjects
Computer simulation, Interface (Java), Computer science, Water flow, Phase (matter), Mathematical analysis, State of art, Compressibility, Point (geometry), Well posedness
Abstract

The modeling and the numerical simulation of two-phase flows are investigated for several decades. When dealing with very heterogeneous problems, for instance a water flow with many bubbles, one has to make use of averaged models since the description of each phase and interface is out of reach. Whatever the average is, the resulting models often suffer from severe mathematical pathologies: lack of hyperbolicity, non-conservative products, non-preservation of admissible states... In 1986, Baer and Nunziato proposed an original model which possesses interesting features from the mathematical point of view. Our goal is to provide a (partial) state of art on this model and its derivatives, but also to list some open questions.

Published
2018
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26. A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles

Author

Nicolas Seguin, Frédéric Coquel, Khaled Saleh, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), 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), Mécanique des Fluides, Energies et Environnement (EDF R&D MFEE), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), 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), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon, Modélisation mathématique, calcul scientifique (MMCS), 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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 Paris-Rocquencourt, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Finite volume method, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Discretization, Applied Mathematics, Numerical analysis, Mathematical analysis, relaxation techniques, Dissipation, Solver, Riemann problem, Riemann solver, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], symbols.namesake, AMS subject classification: 76S05, 35L60, 35F55, Modeling and Simulation, symbols, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Relaxation (approximation), Discontinuous nozzle flows, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is based on a piecewise constant discretization of the cross-section and on an approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous sections with arbitrary large jumps. To do so, we introduce, in the first step of the relaxation solver, a singular dissipation measure superposed on the standing wave, which enables us to control the approximate speeds of sound and thus the time step, even for extreme initial data.

Published
2014
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27. Two properties of two-velocity two-pressure models for two-phase flows

Author

Khaled Saleh, Jean-Marc Hérard, Frédéric Coquel, Nicolas Seguin, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), MFTT, EDF (EDF), 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and The second author received partial support from the NEPTUNE project, which benefits from the financial support of CEA, EDF, AREVA-NP, and IRSN. The last author is partially supported by the LRC Manon (Modélisation et Approximation Numérique Orientées pour l’énergie Nucléaire — CEA DM2S/LJLL).

Subjects
[PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Mechanics of the fluids [physics.class-ph], Applied Mathematics, General Mathematics, Mathematical analysis, Regular polygon, 76T05, 35L60, 35F55, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Classical mechanics, Compressibility, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], symmetrizable system, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], entropy, Two-phase flows, Entropy (arrow of time), Mathematics
Abstract

International audience; We study a class of models of compressible two-phase flows. This class, which includes the Baer-Nunziato model, is based on the assumption that each phase is described by its own pressure, velocity and temperature and on the use of void fractions obtained from averaging process. These models are nonconservative and non-strictly hyperbolic. We prove that the mixture entropy is non-strictly convex and that the system admits a symmetric form.

Published
2014
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28. A finite-volume scheme for a kidney nephron model

Author

Nicolas Seguin, Aurélie Edwards, Magali Tournus, Department of Chemical and Biological Engineering, Tufts University [Medford], 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), Centre de Recherche des Cordeliers (CRC (UMR_S 872)), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Descartes - Paris 5 (UPD5)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Université Paris Descartes - Paris 5 (UPD5)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), 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 ), Centre de Recherche des Cordeliers ( CRC (UMR_S 872) ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National de la Santé et de la Recherche Médicale ( INSERM ) -Centre National de la Recherche Scientifique ( CNRS ), and Tournus, Magali

Subjects
Computer science, 01 natural sciences, 03 medical and health sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Calculus, medicine, QA1-939, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 030304 developmental biology, 0303 health sciences, T57-57.97, Finite volume method, Applied mathematics. Quantitative methods, Courant–Friedrichs–Lewy condition, Mathematical analysis, Stiffness, 010101 applied mathematics, Scheme (mathematics), Transient (oscillation), medicine.symptom, Mathematics, Sign (mathematics), Resolution (algebra)
Abstract

We present a finite volume type scheme to solve a transport nephron model. The model consists in a system of transport equations with specific boundary conditions. The transport velocity is driven by another equation that can undergo sign changes during the transient regime. This is the main difficulty for the numerical resolution. The scheme we propose is based on an explicit resolution and is stable under a CFL condition which does not depend on the stiffness of source terms. Nous présentons un schéma numérique de type volume fini que l’on applique à un modèle de transport dans le néphron. Ce modèle consiste en un système d’équations de transport, avec des conditions aux bords spécifiques. La vitesse du transport est la solution d’un autre système d’équation et peut changer de signe au cours du régime transitoire. Ceci constitue la principale difficulté pour la résolution numérique. Le schéma proposé, basé sur une résolution explicite, est stable sous une condition CFL non restrictive.

Published
2012
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29. Weak solutions to Friedrichs systems with convex constraints

Author

Frédéric Lagoutière, Bruno Després, and Nicolas Seguin

Subjects
Partial differential equation, Applied Mathematics, Existential quantification, Weak solution, Mathematical analysis, Regular polygon, General Physics and Astronomy, Statistical and Nonlinear Physics, Weak formulation, Compact space, Square-integrable function, Applied mathematics, Uniqueness, Mathematical Physics, Mathematics
Abstract

We are interested in a problem arising, for instance, in elastoplasticity modelling, which consists in a system of partial differential equations and a constraint specifying that the solution should remain, for every time and every position, in a certain set. This constraint is generally incompatible with the invariant domains of the original model, thus this problem has to be specified in mathematical terms. Here we follow the approach proposed in Despr?s (2007 Arch. Ration. Mech. Anal. 186 275?308) that furnishes a weak formulation of the constrained problem ? la Kruzhkov. More precisely, this paper deals with the study of the well-posedness of Friedrichs systems under convex constraints, in any space dimension. We prove that there exists a unique weak solution, continuous in time, square integrable in space, and with values in the constraints domain. This is done with the use of a discrete approximation scheme: we define a numerical approximate solution and prove, thanks to compactness properties, that it converges towards a solution to the constrained problem. Uniqueness is proven via energy (or entropy) estimates. Some numerical illustrations are provided.

Published
2011
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30. GODUNOV-TYPE SCHEMES FOR HYPERBOLIC SYSTEMS WITH PARAMETER-DEPENDENT SOURCE: THE CASE OF EULER SYSTEM WITH FRICTION

Author

Pierre-Arnaud Raviart, Frédéric Coquel, Christophe Chalons, Edwige Godlewski, and Nicolas Seguin

Subjects
Applied Mathematics, Mathematical analysis, Type (model theory), Euler system, Solver, Hyperbolic systems, Riemann solver, symbols.namesake, Consistency (statistics), Simple (abstract algebra), Modeling and Simulation, symbols, Parameter dependent, Mathematics
Abstract

Well-balanced or asymptotic preserving schemes are receiving an increasing amount of interest. This paper gives a precise setting for studying both properties in the case of Euler system with friction. We derive a simple solver which, by construction, preserves discrete equilibria and reproduces at the discrete level the same asymptotic behavior as that of the solutions of the continuous system. Numerical illustrations are convincing and show that not all methods share these properties.

Published
2010
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31. The interface coupling of the gas dynamics equations

Author

Christophe Chalons, Pierre-Arnaud Raviart, and Nicolas Seguin

Subjects
symbols.namesake, Coupling (physics), Riemann problem, Uniqueness theorem for Poisson's equation, Applied Mathematics, Weak solution, Mathematical analysis, symbols, Existence theorem, Uniqueness, Compressible flow, Hyperbolic partial differential equation, Mathematics
Abstract

We investigate the one-dimensional coupling of two systems of gas dynamics at a fixed interface. The coupling constraints consist in requiring the continuity of a system of nonconservative variables at the interface. Since we are dealing with hyperbolic systems, weak coupling conditions are proposed. The existence and the uniqueness of the solutions of the coupled Riemann problem are investigated. Several examples of solutions satisfying the weak coupling conditions are contructed, either continuous or discontinuous with respect to the nonconservative variables at the interface.

Published
2008
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32. The drift-flux asymptotic limit of barotropic two-phase two-pressure models

Author

Nicolas Seguin, A. Ambroso, Frédéric Coquel, Thomas Galié, Christophe Chalons, Pierre-Arnaud Raviart, and Edwige Godlewski

Subjects
Physics, drift-flux models, 35C20, 76T10, Applied Mathematics, General Mathematics, Closure (topology), Flux, Mechanics, Type (model theory), asymptotic limit, 35L60, Algebraic closure, two-phase flows, Physics::Fluid Dynamics, Flow (mathematics), Drag, Barotropic fluid, Limit (mathematics)
Abstract

We study the asymptotic behavior of the solutions of barotropic two-phase two-pressure models, with pressure relaxation, drag force and external forces. Using Chapman-Enskog expansions close to the expected equilibrium, a drift-flux model with a Darcy type closure law is obtained. Also, restricting this closure law to permanent flows (defined as steady flows in some Lagrangian frame), we can obtain a drift-flux model with an algebraic closure law, in the spirit of Zuber-Findlay models. The example of a two-phase flow in a vertical pipe is described.

Published
2008
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33. Relaxation methods and coupling procedures

Author

A. Ambroso, Pierre-Arnaud Raviart, Christophe Chalons, Nicolas Seguin, Frédéric Lagoutière, Frédéric Coquel, and Edwige Godlewski

Subjects
Coupling, symbols.namesake, Theoretical physics, Mechanics of Materials, Applied Mathematics, Mechanical Engineering, Computational Mechanics, Euler's formula, symbols, Applied mathematics, Computer Science Applications, Mathematics
Abstract

SUMMARY This paper studies a global relaxation method to ensure the conservative coupling at a fixed interface of two Euler systems with different pressure laws. Copyright q 2007 John Wiley & Sons, Ltd.

Published
2008
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34. Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

Author

Edwige Godlewski, Nicolas Seguin, Clément Cancès, Frédéric Coquel, Hélène Mathis, Reliable numerical approximations of dissipative systems (RAPSODI ), 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, 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), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Numerical Analysis, Geophysics and Ecology (ANGE), 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, LRC MANON (CEA/DM2S -- LJLL) NEEDS program (CNRS -- CEA -- AREVA -- EDF -- IRSN), 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, Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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
Conservation law, Spacetime, Scale (ratio), Applied Mathematics, General Mathematics, Mathematical analysis, 35L65, 35B45, 35B30, 35A35, Flux, 010103 numerical & computational mathematics, Function (mathematics), error estimate, 01 natural sciences, 010101 applied mathematics, Nonlinear system, thick coupling interface, Ordinary differential equation, model adaptation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Hyperbolic partial differential equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Conservation laws
Abstract

International audience; We propose a dynamic model adaptation method for a nonlinear conservation law coupled with an ordinary differential equation. This model, called the ''fine model", involves a small time scale and setting this time scale to 0 leads to a classical conservation law, called the ''coarse model", with a flux which depends on the unknown and on space and time. The dynamic model adaptation consists in detecting the regions where the fine model can be replaced by the coarse one in an automatic way, without deteriorating the accuracy of the result. To do so, we provide an error estimate between the solution of the fine model and the solution of the adaptive method, enabling a sharp control of the different parameters. This estimate rests upon stability results for conservation laws with respect to the flux function. Numerical results are presented at the end and show that our estimate is optimal.

Published
2016
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35. Relaxation approximation of Friedrich's systems under convex constraints

Author

Clément Mifsud, Jean-François Babadjian, Nicolas Seguin, 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

Subjects
Statistics and Probability, Convex analysis, Cauchy problem, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, Regular polygon, Mathematics::Analysis of PDEs, Mathematics::Spectral Theory, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; This paper is devoted to present an approximation of a Cauchy problem for Friedrichs' systems under convex constraints. It is proved the strong convergence in L^2_{loc} of a parabolic-relaxed approximation towards the unique constrained solution.

Published
2016
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36. Error estimate for time-explicit finite volume approximation of strong solutions to systems of conservation laws

Author

Hélène Mathis, Nicolas Seguin, 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 de Mathématiques Jean Leray (LMJL), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Nuclear System and Scenarios federative project of the NEEDS program (CNRS -- CEA -- AREVA -- EDF -- IRSN), LRC Manon (Modélisation et approximation numérique orientées pour l'énergie nucléaire - CEA/DM2S-LJLL)., ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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, Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
Kullback–Leibler divergence, 010103 numerical & computational mathematics, 01 natural sciences, Mathematics - Analysis of PDEs, Approximation error, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Entropy (arrow of time), Mathematics, Numerical Analysis, Conservation law, Finite volume method, Applied Mathematics, Courant–Friedrichs–Lewy condition, Mathematical analysis, relative entropy, 010101 applied mathematics, Computational Mathematics, Nonlinear system, error estimates, finite volume schemes, strong solutions, Round-off error, hyperbolic systems, 35L45, 65M08, 65M12, 65M15, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We study the finite volume approximation of strong solutions to nonlinear systems of conservation laws. We focus on time-explicit schemes on unstructured meshes, with entropy satisfying numerical fluxes. The numerical entropy dissipation is quantified at each interface of the mesh, which enables to prove a weak–BV estimate for the numerical approximation under a strengthen CFL condition. Then we derive error estimates in the multidimensional case, using the relative entropy between the strong solution and its finite volume approximation. The error terms are carefully studied, leading to a classical $h^1/4$ estimate in $L^2$ under this strengthen CFL condition.

Published
2016
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37. ANALYSIS AND APPROXIMATION OF A SCALAR CONSERVATION LAW WITH A FLUX FUNCTION WITH DISCONTINUOUS COEFFICIENTS

Author

Julien Vovelle, Nicolas Seguin, 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), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), 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), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Conservation law, Finite volume method, Applied Mathematics, Numerical analysis, discontinuous coefficient, Mathematical analysis, Godunov's scheme, 010103 numerical & computational mathematics, 01 natural sciences, Nonlinear conservation law, 010101 applied mathematics, resonance, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, finite volume, Entropy (arrow of time), Stationary state, Mathematics
Abstract

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation ut + (k(x)u(1 - u))x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

Published
2003
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38. An energy-consistent depth-averaged Euler system: derivation and properties

Author

Anne Mangeney, Marie-Odile Bristeau, Jacques Sainte-Marie, Nicolas Seguin, 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Physique du Globe de Paris (IPGP), Institut national des sciences de l'Univers (INSU - CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université de La Réunion (UR)-Institut de Physique du Globe de Paris (IPG Paris)-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), Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement - Direction Eau Mer et Fleuves (Cerema Direction Eau Mer et Fleuves), Centre d'Etudes et d'Expertise sur les Risques, l'Environnement, la Mobilité et l'Aménagement (Cerema), Université Pierre et Marie Curie - Paris 6 (UPMC), Centre National de la Recherche Scientifique (CNRS)-Université de La Réunion (UR)-Université Paris Diderot - Paris 7 (UPD7)-IPG PARIS-Institut national des sciences de l'Univers (INSU - CNRS), and Institut national des sciences de l'Univers (INSU - CNRS)-IPG PARIS-Université Paris Diderot - Paris 7 (UPD7)-Université de La Réunion (UR)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Non-hydrostatic model, Free surface flows, Applied Mathematics, Semi-implicit Euler method, Mathematical analysis, Numerical Analysis (math.NA), Euler system, Analytical solutions, Backward Euler method, symbols.namesake, Dispersive terms, Euler's formula, symbols, Compressibility, FOS: Mathematics, Saint-Venant equations, Discrete Mathematics and Combinatorics, Mathematics - Numerical Analysis, Navier-Stokes equations, Asymptotic expansion, Navier–Stokes equations, Shallow water equations, Physics::Atmospheric and Oceanic Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

In this paper, we present an original derivation process of a non-hydrostatic shallow water-type model which aims at approximating the incompressible Euler and Navier-Stokes systems with free surface. The closure relations are obtained by a minimal energy constraint instead of an asymptotic expansion. The model slightly differs from the well-known Green-Naghdi model and is confronted with stationary and analytical solutions of the Euler system corresponding to rotational flows. At the end of the paper, we give time-dependent analytical solutions for the Euler system that are also analytical solutions for the proposed model but that are not solutions of the Green-Naghdi model. We also give and compare analytical solutions of the two non-hydrostatic shallow water models.

Published
2015

39. Dynamic model adaptation for multiscale simulation of hyperbolic systems with relaxation

Author

Nicolas Seguin, Hélène Mathis, Clément Cancès, Edwige Godlewski, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Reliable numerical approximations of dissipative systems (RAPSODI ), 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, 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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 Paris-Rocquencourt, LRC Manon (Modélisation et approximation numérique orientées pour l'énergie nucléaire - CEA/DM2S-LJLL)., Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), 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, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)

Subjects
finite volume methods, 010103 numerical & computational mathematics, Computational fluid dynamics, 35L45, 65M08, 65M55, 35C20, 76T10, 01 natural sciences, two-phase flows, Theoretical Computer Science, relaxation, Hyperbolic system, dynamic model adaptation, multiscale method, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Chapman-Enskog expansion, Spurious relationship, Hyperbolic equilibrium point, Mathematics, Numerical Analysis, Finite volume method, business.industry, Applied Mathematics, Numerical analysis, Mathematical analysis, General Engineering, Hyperbolic systems, 010101 applied mathematics, Computational Mathematics, Test case, Computational Theory and Mathematics, A priori and a posteriori, business, Software, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In numerous industrial CFD applications, it is usual to use two (or more)different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding hyperbolic equilibrium systems. Using a numerical Chapman-Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.

Published
2015
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40. Some recent finite volume schemes to compute Euler equations using real gas EOS

Author

Jean-Marc Hérard, Nicolas Seguin, and Thierry Gallouët

Subjects
Shock wave, Finite volume method, Real gas, Applied Mathematics, Mechanical Engineering, Mathematical analysis, Computational Mechanics, 010103 numerical & computational mathematics, Perfect gas, 01 natural sciences, Computer Science Applications, Euler equations, 010101 applied mathematics, Discontinuity (linguistics), symbols.namesake, Mechanics of Materials, symbols, Sod shock tube, 0101 mathematics, Shock tube, Mathematics, Mathematical physics
Abstract

This paper deals with the resolution by finite volume methods of Euler equations in one space dimension, with real gas state laws (namely, perfect gas EOS, Tammann EOS and Van Der Waals EOS). All tests are of unsteady shock tube type, in order to examine a wide class of solutions, involving Sod shock tube, stationary shock wave, simple contact discontinuity, occurrence of vacuum by double rarefaction wave, propagation of a one-rarefaction wave over ‘vacuum’, … Most of the methods computed herein are approximate Godunov solvers: VFRoe, VFFC, VFRoe ncv (τ, u, p) and PVRS. The energy relaxation method with VFRoe ncv (τ, u, p) and Rusanov scheme have been investigated too. Qualitative results are presented or commented for all test cases and numerical rates of convergence on some test cases have been measured for first- and second-order (Runge–Kutta 2 with MUSCL reconstruction) approximations. Note that rates are measured on solutions involving discontinuities, in order to estimate the loss of accuracy due to these discontinuities. Copyright © 2002 John Wiley & Sons, Ltd.

Published
2002
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41. A Robust Entropy-Satisfying Finite Volume Scheme for the Isentropic Baer-Nunziato Model

Author

Nicolas Seguin, Jean-Marc Hérard, Frédéric Coquel, Khaled Saleh, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Mécanique des Fluides, Energies et Environnement (EDF R&D MFEE), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), EDF (EDF), 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), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), 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), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon, Modélisation mathématique, calcul scientifique (MMCS), 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
Numerical Analysis, Finite volume method, Entropy (statistical thermodynamics), Applied Mathematics, Degenerate energy levels, Mathematical analysis, Solver, relaxation techniques, AMS subject classifications: 76T05, 35L60, 35F55, Riemann problem, Riemann solver, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Computational Mathematics, symbols.namesake, Riemann hypothesis, Modeling and Simulation, Riemann sum, symbols, entropy-satisfying methods, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Two-phase flows, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

42 pages; We construct an approximate Riemann solver for the isentropic Baer-Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.

Published
2014
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42. Well-posedness for a one-dimensional fluid-particle interaction model

Author

Frédéric Lagoutière, Takéo Takahashi, Boris Andreianov, Nicolas Seguin, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), 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), Numerical Analysis, Geophysics and Ecology (ANGE), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Robust control of infinite dimensional systems and applications (CORIDA), Institut Élie Cartan de Nancy (IECN), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Lorraine (INPL)-Université Nancy 2-Université Henri Poincaré - Nancy 1 (UHP)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), ANR-11-JS01-0006,CoToCoLa,Thématiques actuelles en lois de conservation(2011), Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Seguin, Nicolas, and Jeunes Chercheuses et Jeunes Chercheurs - Thématiques actuelles en lois de conservation - - CoToCoLa2011 - ANR-11-JS01-0006 - JCJC - VALID

Subjects
Conservation law, Differential equation, Point particle, Applied Mathematics, Mathematical analysis, Ode, 35L65, 35L81, 35R06, 65M12, Fixed point, Non-conservative coupling, Burgers' equation, Burgers equation, Computational Mathematics, Splitting, Well-posedness, Wave-front tracking, Fixed-Point, Bounded variation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Entropy (arrow of time), BV estimates, Analysis, Fluid-particle interaction, Mathematics
Abstract

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L∞ data is also given.

Published
2014

43. A simple derivation of BV bounds for inhomogeneous relaxation systems

Author

Nicolas Seguin, Magali Tournus, Benoît Perthame, 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), Modelling and Analysis for Medical and Biological Applications (MAMBA), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Pierre et Marie Curie - Paris 6 (UPMC), Numerical Analysis, Geophysics and Ecology (ANGE), Department of Mathematics [Philadelphia], University of Pennsylvania, and University of Pennsylvania [Philadelphia]

Subjects
Conservation law, Applied Mathematics, General Mathematics, strong compactness, spatial heterogeneity, 010103 numerical & computational mathematics, entropy condition, 01 natural sciences, 35L03, 35L60, 35B40, 35Q92, 010101 applied mathematics, Mathematics - Analysis of PDEs, Compact space, Hyperbolic relaxation, boundary conditions, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Heterogeneous source, Entropy (arrow of time), Mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; We consider relaxation systems of transport equations with heterogeneous source terms and with boundary conditions, which limits are scalar conservation laws. Classical bounds fail in this context and in particular BV estimates. They are the most standard and simplest way to prove compactness and convergence. We provide a novel and simple method to obtain partial BV regularity and strong compactness in this framework. The standard notion of entropy is not convenient either and we also indicate another, but closely related, notion. We give two examples motivated by renal flows which consist of 2 by 2 and 3 by 3 relaxation systems with 2-velocities but the method is more general.

Published
2014
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44. Petiroc, a new front-end ASIC for time of flight application

Author

J. Fleury, Stéphane Callier, G. W. Martin, S. Ahmad, Nicolas Seguin, C. De La Taille, F. Dulucq, and D. Thienpont

Subjects
Physics, Time of flight, Silicon photomultiplier, Optics, Application-specific integrated circuit, business.industry, Electronic engineering, Linearity, High voltage, business, Noise (electronics), Dark current, Jitter
Abstract

Petiroc is a 16-channel front-end ASIC designed to readout silicon photomultipliers (SiPMs) for particle time-of-flight measurement applications. Petiroc combines a very fast and low-jitter trigger with an accurate charge measurement. The concept of the ASIC is to combine two measurement lines that won't interfere one with another to measure both first incident photon timing measurement and whole crystal light charge integration. An adjustment of the SiPM high voltage is possible using a channel-by-channel input DAC. That allows a fine SiPM gain and dark noise adjustment at the system level to correct for the non-uniformity of SiPMs. The power consumption is 3.5 mW/channel, excluding ASIC outing buffer. First measurement on Petiroc shows a time jitter down to 16ps on 20 photoelectrons test pulses and 46ps with 15 photoelectrons from a Hamamatsu MPPC. Charge measurement has been measured and 1% linearity has been measured up to 2000 photoelectrons. Energy resolution has been measured at 9.5% FWMH. The main application of Petiroc is PET time-of-flight prototyping but can be used for any application that requires both sharp time resolution and precise energy measurement.

Published
2013
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45. MODELLING COMPRESSIBLE MULTIPHASE FLOWS

Author

Nicolas Seguin, Thierry Gallouët, Jean-Marc Hérard, Philippe Helluy, Olivier Hurisse, Frédéric Coquel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), EDF (EDF), 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), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est

Subjects
Operations research, Computer simulation, Computer science, Multiphase flow, Mechanics, 01 natural sciences, 010305 fluids & plasmas, 010101 applied mathematics, Physics::Fluid Dynamics, Closure (computer programming), Flow (mathematics), 0103 physical sciences, Compressibility, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [MATH]Mathematics [math]
Abstract

International audience; We give in this paper a short review of some recent achievements within the framework of multiphase flow modeling. We focus first on a class of compressible two-phase flow models, detailing closure laws and their main properties. Next we briefly summarize some attempts to model two-phase flows in a porous region, and also a class of compressible three-phase flow models. Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are then discussed, and we eventually show some numerical results when tackling two-phase flows with mass transfer.

Published
2013
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46. A model of calcium transport along the rat nephron

Author

Benoît Perthame, Magali Tournus, Aurélie Edwards, Nicolas Seguin, S. Randall Thomas, 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), Centre de Recherche des Cordeliers (CRC (UMR_S 872)), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Descartes - Paris 5 (UPD5)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC), Numerical Analysis, Geophysics and Ecology (ANGE), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Nonlinear Analysis for Biology and Geophysical flows (BANG), Unite de recherche en résonance magnétique médicale (U2R2M), Université Paris-Sud - Paris 11 (UP11)-Hôpital Bicêtre-Centre National de la Recherche Scientifique (CNRS), UPMC-EME0918, Emergence 0918, Université Paris Descartes - Paris 5 (UPD5)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), 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 ), Centre de Recherche des Cordeliers ( CRC (UMR_S 872) ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National de la Santé et de la Recherche Médicale ( INSERM ) -Centre National de la Recherche Scientifique ( CNRS ), Université Pierre et Marie Curie - Paris 6 ( UPMC ), Numerical Analysis, Geophysics and Ecology ( ANGE ), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Nonlinear Analysis for Biology and Geophysical flows ( BANG ), Unite de recherche en résonance magnétique médicale ( U2R2M ), and Université Paris-Sud - Paris 11 ( UP11 ) -Hôpital Bicêtre-Centre National de la Recherche Scientifique ( CNRS )

Subjects
kidney, medicine.medical_specialty, Medullary cavity, Physiology, media_common.quotation_subject, [SDV]Life Sciences [q-bio], 030232 urology & nephrology, chemistry.chemical_element, Biological Transport, Active, Nephron, Calcium, Biochemistry, Models, Biological, Marie curie, 03 medical and health sciences, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0302 clinical medicine, Internal medicine, Genetics, medicine, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Animals, Hypercalciuria, Molecular Biology, media_common, 030304 developmental biology, 0303 health sciences, Kidney, [ SDV ] Life Sciences [q-bio], urogenital system, Chemistry, Osmolar Concentration, Art, Anatomy, Nephrons, medicine.disease, Rats, medicine.anatomical_structure, Endocrinology, transport, Models, Animal, Humanities, finite volume, mathematical model, Biotechnology
Abstract

We developed a mathematical model of calcium (Ca2+) transport along the rat nephron to investigate the factors that promote hypercalciuria. The model is an extension of the flat medullary model of Hervy and Thomas ( Am J Physiol Renal Physiol 284: F65–F81, 2003). It explicitly represents all the nephron segments beyond the proximal tubules and distinguishes between superficial and deep nephrons. It solves dynamic conservation equations to determine NaCl, urea, and Ca2+concentration profiles in tubules, vasa recta, and the interstitium. Calcium is known to be reabsorbed passively in the thick ascending limbs and actively in the distal convoluted (DCT) and connecting (CNT) tubules. Our model predicts that the passive diffusion of Ca2+from the vasa recta and loops of Henle generates a significant axial Ca2+concentration gradient in the medullary interstitium. In the base case, the urinary Ca2+concentration and fractional excretion are predicted as 2.7 mM and 0.32%, respectively. Urinary Ca2+excretion is found to be strongly modulated by water and NaCl reabsorption along the nephron. Our simulations also suggest that Ca2+molar flow and concentration profiles differ significantly between superficial and deep nephrons, such that the latter deliver less Ca2+to the collecting duct. Finally, our results suggest that the DCT and CNT can act to counteract upstream variations in Ca2+transport but not always sufficiently to prevent hypercalciuria.

Published
2013
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47. General constrained conservation laws. Application to pedestrian flow modeling

Author

Paola Goatin, Nicolas Seguin, Christophe Chalons, 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 Paris Diderot - Paris 7 (UPD7), Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE (OPALE), 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 (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), Université Pierre et Marie Curie - Paris 6 (UPMC), European Project: 257661,EC:FP7:ERC,ERC-2010-StG_20091028,TRAM3(2010), 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), and 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)

Subjects
Statistics and Probability, Conservation law, Finite volume method, Applied Mathematics, Mathematical analysis, General Engineering, 010103 numerical & computational mathematics, Pedestrian flow, Classification of discontinuities, 01 natural sciences, Riemann solver, Computer Science Applications, 010101 applied mathematics, symbols.namesake, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics
Abstract

International audience; We generalize the results on conservation laws with local flux constraint obtained in [1, 9] to general flux functions and nonclassical solutions arising for example in pedestrian flow modeling. We first define the constrained Riemann solver and the entropy condition, which singles out the unique admissible solution. We provide a well posedness result based on wave-front tracking approximations and Kruzhkov doubling of variable technique. We then provide the framework to deal with nonclassical solutions and we propose a "front-tracking" finite volume scheme allowing to sharply capture classical and nonclassical discontinuities. Numerical simulations illustrating the Braess paradox are presented as validation of the method.

Published
2013

48. Foreword

Author

Fré́dé́ric Coquel, Michaël Gutnic, Philippe Helluy, Fré́dé́ric Lagoutière, Christian Rohde, and Nicolas Seguin

Subjects
T57-57.97, Applied mathematics. Quantitative methods, QA1-939, Mathematics
Published
2013

49. OSAMOAL: optimized simulations by adapted models using asymptotic limits

Author

Hélène Mathis, Khaled Saleh, Nicolas Seguin, Clément Cancès, Anne-Céline Boulanger, Nonlinear Analysis for Biology and Geophysical flows (BANG), 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 Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), 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, Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), 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), Université Claude Bernard Lyon 1 (UCBL), Université de Lyon, Modélisation mathématique, calcul scientifique (MMCS), 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), Project of the Cemracs 2011, supported by the LRC Manon (Modélisation et Approximation Numérique Orientées pour l'énergie Nucléaire - CEA/DM2S-LJLL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

Subjects
T57-57.97, Applied mathematics. Quantitative methods, MathematicsofComputing_NUMERICALANALYSIS, 010103 numerical & computational mathematics, 01 natural sciences, parabolic limit, Hyperbolic systems, 010101 applied mathematics, Calculus, QA1-939, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, hyperbolic systems, asymptotic preserving schemes, interfacial coupling, Mathematics
Abstract

We propose in this work to address the problem of model adaptation, dedicated to hyperbolic models with relaxation and to their parabolic limit. The goal is to replace a hyperbolic system of balance laws (the so-called fine model) by its parabolic limit (the so-called coarse model), in delimited parts of the computational domain. Our method is based on the construction of asymptotic preserving schemes and on interfacial coupling methods between hyperbolic and parabolic models. We study in parallel the cases of the Goldstein-Taylor model and of the p-system with friction. Nous proposons dans ce travail de traiter le problème d’adaptation de modèle, appliqué aux systèmes hyperboliques de relaxation et à leur limite parabolique. Le but est de remplacer dans des zones délimitées du domaine de calcul un système hyperbolique avec terme source (le modèle fin) par le modèle parabolique limite associé (le modèle grossier). Notre méthode repose sur des schémas préservant cette asymptotique et le couplage interfacial entre des modèles hyperbolique et parabolique. On étudie les cas du modèle de Goldstein-Taylor et du p-système avec friction avec leurs limites paraboliques respectives.

Published
2012
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50. A Class of Two-fluid Two-phase Flow Models

Author

Nicolas Seguin, Frédéric Coquel, Jean-Marc Hérard, Khaled Saleh, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), EDF (EDF), 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), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), 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), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and 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)

Subjects
Convection, Class (set theory), Approximations of π, 010103 numerical & computational mathematics, 16. Peace & justice, 01 natural sciences, Physics::Fluid Dynamics, 010101 applied mathematics, Flow (mathematics), Jump, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Order (group theory), Applied mathematics, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Two-phase flow, Uniqueness, 0101 mathematics, Mathematics
Abstract

International audience; We introduce a class of two-fluid models that complies with a few theoretical requirements that include: (i) hyperbolicity of the convective subset, (ii) entropy inequality, (iii) uniqueness of jump conditions for non-viscous flows. These specifications are necessary in order to compute relevant approximations of unsteady flow patterns. It is shown that the Baer-Nunziato model belongs to this class of two-phase flow models, and the main properties of the model are given, before showing a few numerical experiments.

Published
2012
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