Author: "Fré́dé́ric Lagoutière" - Searchworks@Jio Institute Digital Library Search Results

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67 results on '"Fré́dé́ric Lagoutière"'

1. Relaxation Limit of the Aggregation Equation with Pointy Potential

Author: Benoît Fabrèges, Frédéric Lagoutière, Sébastien Tran Tien, and Nicolas Vauchelet
Subjects: aggregation equation, relaxation limit, scalar conservation law, finite volume scheme, Mathematics, QA1-939
Abstract: This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes.
Published: 2021
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2. Mathematical justification of a compressible bi-fluid system with different pressure laws: A semi-discrete approach and numerical illustrations.

Author: Didier Bresch, Cosmin Burtea, and Frédéric Lagoutière
Published: 2023
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3. Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle.

Author: Nina Aguillon, Frédéric Lagoutière, and Nicolas Seguin
Published: 2017
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4. Mathematical justification of a compressible bifluid system with different pressure laws: a continuous approach

Author: Didier Bresch, Cosmin Burtea, and Frédéric Lagoutière
Subjects: Applied Mathematics, Analysis
Published: 2022
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5. Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model.

Author: Boris P. Andreianov, Frédéric Lagoutière, Nicolas Seguin, and Takéo Takahashi
Published: 2014
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6. An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model.

Author: Samuel Kokh and Frédéric Lagoutière
Published: 2010
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7. Small solids in an inviscid fluid.

Author: Boris P. Andreianov, Frédéric Lagoutière, Nicolas Seguin, and Takéo Takahashi
Published: 2010
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8. Coupling of general Lagrangian systems.

Author: A. Ambroso, Christophe Chalons, Frédéric Coquel, Edwige Godlewski, Frédéric Lagoutière, Pierre-Arnaud Raviart, and Nicolas Seguin
Published: 2008
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9. Genuinely Multi-Dimensional Non-Dissipative Finite-Volume Schemes for Transport.

Author: Bruno Després and Frédéric Lagoutière
Published: 2007
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10. Contact Discontinuity Capturing Schemes for Linear Advection and Compressible Gas Dynamics.

Author: Bruno Després and Frédéric Lagoutière
Published: 2001
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11. The Neumann numerical boundary condition for transport equations

Author: Frédéric Lagoutière, Jean-François Coulombel, Institut de Mathématiques de Toulouse UMR5219 (IMT), 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), 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), 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), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), Centre National de la Recherche Scientifique ( CNRS ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -École Centrale de Lyon ( ECL ), Université de Lyon-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 Claude Bernard Lyon 1 ( UCBL ), Université de Lyon, ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion. ( 2013 ), ANR-17-CE40-0025,NABUCO,NumericAl BoUndaries and COupling, 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), 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), 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: Discretization, Type (model theory), 01 natural sciences, Stability (probability), 010305 fluids & plasmas, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Normal mode, numerical schemes, 0103 physical sciences, Convergence (routing), Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, convergence, Mathematical analysis, transport equations, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, 010101 applied mathematics, Modeling and Simulation, Outflow boundary, AMS classification: 65M12, 65M06, 65M20, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in \begin{document}$ \ell^\infty $\end{document} for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.
Published: 2020
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12. Special issue on contemporary topics in conservation laws.

Author: Rinaldo M. Colombo, Kenneth H. Karlsen, Frédéric Lagoutière, and Andrea Marson
Published: 2016
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13. Relaxation limit of the aggregation equation with pointy potential

Author: Nicolas Vauchelet, Benoît Fabrèges, Tran Tien, Frédéric Lagoutière, 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-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), Laboratoire Analyse, Géométrie et Applications (LAGA), and Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord
Subjects: Work (thermodynamics), Relaxation limit, Discretization, Logic, One-dimensional space, 010103 numerical & computational mathematics, 01 natural sciences, Aggregation equation, Mathematics - Analysis of PDEs, Dimension (vector space), Convergence (routing), FOS: Mathematics, QA1-939, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Scalar conservation law, Mathematical Physics, Mathematics, Algebra and Number Theory, Newtonian potential, 010102 general mathematics, Finite volume scheme, Geometry and Topology, Relaxation (approximation), Analysis, Analysis of PDEs (math.AP)
Abstract: This work was devoted to the study of a relaxation limit of the so-called aggregation equation with a pointy potential in one-dimensional space. The aggregation equation is today widely used to model the dynamics of a density of individuals attracting each other through a potential. When this potential is pointy, solutions are known to blow up in final time. For this reason, measure-valued solutions have been defined. In this paper, we investigated an approximation of such measure-valued solutions thanks to a relaxation limit in the spirit of Jin and Xin. We study the convergence of this approximation and give a rigorous estimate of the speed of convergence in one dimension with the Newtonian potential. We also investigated the numerical discretization of this relaxation limit by uniformly accurate schemes.
Published: 2021

14. Error estimates of finite difference schemes for the Korteweg–de Vries equation

Author: Clémentine Courtès, Frédéric Lagoutière, and Frédéric Rousset
Subjects: Pure mathematics, Applied Mathematics, General Mathematics, Numerical analysis, Finite difference, Order (ring theory), 010103 numerical & computational mathematics, 01 natural sciences, 010101 applied mathematics, Sobolev space, Computational Mathematics, Rate of convergence, Convergence (routing), Initial value problem, 0101 mathematics, Korteweg–de Vries equation, Mathematics
Abstract: This article deals with the numerical analysis of the Cauchy problem for the Korteweg–de Vries equation with a finite difference scheme. We consider the explicit Rusanov scheme for the hyperbolic flux term and a 4-point $\theta $-scheme for the dispersive term. We prove the convergence under a hyperbolic Courant–Friedrichs–Lewy condition when $\theta \geq \frac{1}{2}$ and under an ‘Airy’ Courant–Friedrichs–Lewy condition when $\theta
Published: 2018
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15. Convergence order of upwind type schemes for transport equations with discontinuous coefficients

Author: Frédéric Lagoutière, Nicolas Vauchelet, François Delarue, 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), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-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), 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), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)
Subjects: General Mathematics, conservative transport equation, Upwind scheme, 010103 numerical & computational mathematics, 01 natural sciences, Mathematics - Analysis of PDEs, upwind finite volume scheme, Convergence (routing), FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, convergence order, Mathematics, Finite volume method, Applied Mathematics, continuity equation, 010102 general mathematics, Numerical Analysis (math.NA), 16. Peace & justice, Lipschitz continuity, Rate of convergence, Continuity equation, Bounded function, measure-valued solution, 2010 AMS: 35D30, 35L65, 65M12, 65M15, Vector field, forward semi-Lagrangian scheme, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the upwind scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other " diffusive " " first order " schemes and to a forward semi-Lagrangian scheme.
Published: 2017
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16. Numerical investigations of the compressible Navier-Stokes system

Author: Thierry Goudon, Frédéric Lagoutière, Jean-Jéróme Casanova, Sebastian Minjeaud, Pauline Lafitte, Bilal Al-Taki, Kevin Atsou, Peking University [Beijing], COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Mathématiques et Informatique pour la Complexité et les Systèmes (MICS), CentraleSupélec-Université Paris-Saclay, 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), 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 Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), 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), Modélisation mathématique, calcul scientifique (MMCS), 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)-École Centrale de Lyon (ECL)
Subjects: T57-57.97, Applied mathematics. Quantitative methods, 010102 general mathematics, Linear system, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Eulerian path, 010103 numerical & computational mathematics, 01 natural sciences, Projection (linear algebra), symbols.namesake, Nonlinear system, Discontinuity (linguistics), Inviscid flow, Barotropic fluid, symbols, Euler's formula, QA1-939, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH]Mathematics [math], Mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: In this paper we write, analyze and experimentally compare three different numerical schemes dedicated to the one dimensional barotropic Navier-Stokes equations: a staggered scheme based on the Rusanov one for the inviscid (Euler) system,a staggered pseudo-Lagrangian scheme in which the mesh “follows” the fluid,the Eulerian projection (on a fixed mesh) of the preceding scheme. All these schemes only involve the resolution of linear systems (all the nonlinear terms are solved in an explicit way). We propose numerical illustrations of their behaviors on particular solutions in which the density has discontinuities (hereafter called Hoff solutions). We show that the three schemes seem to converge to the same solutions, and we compare the evolution of the amplitude of the discontinuity of the numerical solution (with the pseudo-Lagrangian scheme) with the one predicted by Hoff and observe a good agreement.
Published: 2020
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17. A New Numerical algorithm for two-phase flows drift-flux model with staggered grid in porous media

Author: Stéphane Dellacherie, Gloria Faccanoni, Anne Charmeau, Anouar Mekkas, Bérénice Grec, Frédéric Lagoutière, Sami Kouraichi, Yohan Penel, 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, and Université Paris 13 (UP13)
Subjects: Shock wave, T57-57.97, Finite volume method, Applied mathematics. Quantitative methods, [PHYS.NUCL]Physics [physics]/Nuclear Theory [nucl-th], Numerical analysis, [PHYS.NEXP]Physics [physics]/Nuclear Experiment [nucl-ex], Solver, Grid, symbols.namesake, Mach number, Pressure-correction method, Compressibility, symbols, QA1-939, Algorithm, Mathematics
Abstract: FLICA4 is a 3D compressible code dedicated to reactor core analysis. FLICA4 solves for two-phase flows in porous media using a compressible drift-flux model [2]. To derive convective fluxes, FLICA4 uses a specific finite volume numerical method based on an extension of the Roe’s approximate Riemann colocated solver [3]. The method is developped to solve 1D flows. It requires adjustments to solve for 2D or 3D flows, at low Mach number, on a cartesian mesh. As Mach number goes to zero, in some extreme 2D or 3D cases, the native method does converge but the solution is not consistent with the analytical or experimental solution [4]. A so-called “pressure correction“ is added to the initial solver to derive an accurate solution. However, this extra term produces checkerboard oscillations in space for most applications of interest, even when solving for 1D flows. These checkerboard oscillations are sometimes critical and may lead to unstable solutions or divergence of the solver. The paper presents the analysis of an alternative numerical algorithm to solve for compressible drift-flux model in the low Mach regime. The current study aims at investigating a new compressible scheme that is both accurate and robust at low Mach number on staggered grids. The actual behavior of the new scheme is tested against simplified nuclear core ”boiling channel” study cases in the low Mach regime. The tests are extended to compressible regime for flows exhibiting both shock waves and no shock waves. The paper shows that the alternative numerical algorithm leads to numerical solutions that are very close to analytical solutions. Moreover, no more checkboard oscillations are generated. Therefore, the algorithm is a real improvement to FLICA4 current numerical method.
Published: 2017
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18. Modeling and analysis of adipocytes dynamic with a differentiation process

Author: Thierry Goudon, Benjamin Mauroy, Pascal Millet, Frédéric Lagoutière, Jérôme Gilleron, Hugo Martin, Cristina Vaghi, Magali Ribot, Centre méditerranéen de médecine moléculaire (C3M), 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)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Université Côte d'Azur (UCA), COmplex Flows For Energy and Environment (COFFEE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), 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), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Université de Lyon, Université Claude Bernard Lyon 1 (UCBL), 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), Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jean Alexandre Dieudonné (LJAD), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Modélisation Mathématique pour l'Oncologie (MONC), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], UNICANCER-UNICANCER-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Vaghi, Cristina, 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), Institut Camille Jordan [Villeurbanne] (ICJ), Laboratoire Jean Alexandre Dieudonné (JAD), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)-Institut National de la Santé et de la Recherche Médicale (INSERM), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), 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 normale supérieure - Paris (ENS Paris), Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Institut Bergonié [Bordeaux], Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), and Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO)
Subjects: 0301 basic medicine, Surface (mathematics), education.field_of_study, T57-57.97, Applied mathematics. Quantitative methods, Advection, Population, 030209 endocrinology & metabolism, Radius, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, 03 medical and health sciences, 030104 developmental biology, 0302 clinical medicine, Simple (abstract algebra), Scientific method, Ordinary differential equation, QA1-939, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, [INFO.INFO-MO] Computer Science [cs]/Modeling and Simulation, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], education, Mathematics
Abstract: We propose in this article a model describing the dynamic of a system of adipocytes, struc-tured by their sizes. This model takes into account the differentiation of a population of mesenchymal cells into preadipocytes and of preadipocytes into adipocytes; the differentiation rates depend on the mean adipocyte radius. The considered equations are therefore ordinary differential equations, coupled with an advection equation, the growth rate of which depends on food availability and on the total surface of adipocytes. Since this velocity is discontinuous, we need to introduce a convenient notion of solutions coming from Filippov theory. We are consequently able to determine the stationary solutions of the system, to prove the existence and uniqueness of solutions and to describe the asymptotic behavior of solutions in some simple cases. Finally, the parameters of the model are fitted thanks to some experimental data and numerical simulations are displayed; a spatial extension of the model is studied numerically., Nous proposons dans cet article un modèle décrivant la dynamique d'un ensemble d'adipocy-tes structurés en taille. Ce modèle tient compte de la différentiation d'une population de cellules mésenchymales en préadipocytes et des préadipocytes en adipocytes; les taux de différentiation dépen-dent du rayon moyen des adipocytes. LeséquationsLeséquations considérées sont, par conséquence, deséquationsdeséquations différentielles ordinaires, couplées avec uné equation d'advection, dont le taux de croissance dépend de la nourriture disponible et de la surface totale des adipocytes. Comme la vitesse est discontinue, nous introduisons une notion appropriée des solutions, provenant de la théorie de Filippov. Nous sommes alors en mesure de déterminer les solutions stationnaires du système, de prouver l'existence et l'unicité de solutions et de décrire le comportement asymptotique des solutions dans certains cas simples. Finalement, les paramètres du modèles sont estimés grâcè a des données expérimentales et des simulations numériques sont présentées; une extension spatiale du modèles estétudiéeestétudiée numériquement.
Published: 2019
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19. Numerical schemes for the aggregation equation with pointy potentials

Author: François Delarue, Sofiane Martel, Frédéric Lagoutière, Kévin Le Balc'h, Hélène Hivert, Nicolas Vauchelet, Benoît Fabrèges, 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), Modélisation mathématique, calcul scientifique (MMCS), 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), 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), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (Saint-Venant), 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), 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), 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), 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-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 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 ), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics ( Saint-Venant ), 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 ), Laboratoire Jean Alexandre Dieudonné ( JAD ), Université Nice Sophia Antipolis ( UNS ), Université Côte d'Azur ( UCA ) -Université Côte d'Azur ( UCA ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Analyse, Géométrie et Applications ( LAGA ), Université Paris 8 Vincennes-Saint-Denis ( UP8 ) -Université Paris 13 ( UP13 ) -Université Sorbonne Paris Cité ( USPC ) -Institut Galilée-Centre National de la Recherche Scientifique ( CNRS ), 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), 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 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), Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics (LHSV), 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 Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)
Subjects: T57-57.97, Finite volume method, Applied mathematics. Quantitative methods, Collective motion, 02 engineering and technology, 01 natural sciences, Measure (mathematics), Nonlinear conservation law, 010101 applied mathematics, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Convergence (routing), 0202 electrical engineering, electronic engineering, information engineering, QA1-939, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 020201 artificial intelligence & image processing, 0101 mathematics, Finite time, Focus (optics), Mathematics
Abstract: International audience; The aggregation equation is a nonlocal and nonlinear conservation law commonly used to describe the collective motion of individuals interacting together. When interacting potentials are pointy, it is now well established that solutions may blow up in finite time but global in time weak measure valued solutions exist. In this paper we focus on the convergence of particle schemes and finite volume schemes towards these weak measure valued solutions of the aggregation equation.
Published: 2019
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20. Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions

Author: Hans Henrik Rugh, Mélanie Inglard, Frédéric Lagoutière, 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-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), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)
Subjects: 010101 applied mathematics, Mathematical analysis, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010103 numerical & computational mathematics, General Medicine, 0101 mathematics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: International audience
Published: 2019

21. SIMULATIONS OF THE LIFSHITZ–SLYOZOV EQUATIONS: THE ROLE OF COAGULATION TERMS IN THE ASYMPTOTIC BEHAVIOR

Author: Léon Matar Tine, Frédéric Lagoutière, and Thierry Goudon
Subjects: Finite volume method, Applied Mathematics, Modeling and Simulation, Coagulation (water treatment), Statistical physics, Numerical diffusion, Mathematics
Abstract: We consider the Lifshitz–Slyozov system that describes the kinetics of precipitation from supersaturated solid solutions. We design specific Finite Volume schemes and we investigate numerically the behavior of the solutions, in particular the large time asymptotics. Our purpose is two-fold: first, we introduce an adapted scheme based on downwinding techniques in order to reduce the numerical diffusion; second, we discuss the influence of coagulation effects on the selection of the asymptotic profile.
Published: 2013
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22. Convergence analysis of upwind type schemes for the aggregation equation with pointy potential

Author: Nicolas Vauchelet, Frédéric Lagoutière, François Delarue, 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), 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), 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), 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), BQR Accueil EC 2017, and ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013)
Subjects: 35B40, 35D30, 35L60, 35Q92, 49K20, Finite volume method, Numerical analysis, 010102 general mathematics, Ocean Engineering, 01 natural sciences, aggregation equation, 010101 applied mathematics, Nonlinear system, Mathematics - Analysis of PDEs, Rate of convergence, Flow (mathematics), upwind finite volume scheme, Convergence (routing), FOS: Mathematics, measure-valued solution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Uniqueness, 0101 mathematics, convergence order, Convection–diffusion equation, Mathematics, Analysis of PDEs (math.AP)
Abstract: International audience; A numerical analysis of upwind type schemes for the nonlinear nonlocal aggregation equation is provided. In this approach, the aggregation equation is interpreted as a conservative transport equation driven by a nonlocal nonlinear velocity field with low regularity. In particular, we allow the interacting potential to be pointy, in which case the velocity field may have discontinuities. Based on recent results of existence and uniqueness of a Filippov flow for this type of equations, we study an upwind finite volume numerical scheme and we prove that it is convergent at order 1/2 in Wasserstein distance. This result is illustrated by numerical simulations, which show the optimality of the order of convergence.
Published: 2017
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23. Analysis and simulation of nonlinear and nonlocal transport equations

Author: Frédéric Lagoutière, Nicolas Vauchelet, Laboratoire de Mathématiques d'Orsay (LMO), Université Paris-Sud - Paris 11 (UP11)-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), 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 de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013), Lagoutière, Frédéric, and Blanc 2013 - Modèles cinétiques en biologie et domaines connexes - - KIBORD2013 - ANR-13-BS01-0004 - Blanc 2013 - VALID
Subjects: Computer simulation, numerical analysis, analysis, Numerical analysis, 010102 general mathematics, Mathematical analysis, 010103 numerical & computational mathematics, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Measure (mathematics), Pushforward measure, blowup, Nonlinear system, Aggregation equation, Initial value problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Vector field, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Convection–diffusion equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, existence and uniqueness
Abstract: This article is devoted to the analysis of some nonlinear conservative transport equations, including the so-called aggregation equation with pointy potential, and numerical method devoted to its numerical simulation. Such a model describes the collective motion of individuals submitted to an attractive potential and can be written as a continuity transport equation with a velocity field computed through a self-consistent interaction potential. In the strongly attractive setting, L p solutions may blow up in finite time, then a theory of existence of weak measure solutions has been defined. In this approach, we show the existence of Filippov characteristics allowing to define solutions of the aggregation initial value problem as a pushforward measure. Then numerical analysis of an upwind type scheme is proposed allowing to recover the dynamics of aggregates beyond the blowup time.
Published: 2016

24. 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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25. Probabilistic Analysis of the Upwind Scheme for Transport Equations

Author: François Delarue and Frédéric Lagoutière
Subjects: Markov chain, Mechanical Engineering, Geodetic datum, Upwind scheme, 010103 numerical & computational mathematics, Numerical diffusion, Lipschitz continuity, 01 natural sciences, 010101 applied mathematics, Mathematics (miscellaneous), Flow (mathematics), Applied mathematics, Probabilistic analysis of algorithms, 0101 mathematics, Representation (mathematics), Analysis, Mathematics
Abstract: We provide a probabilistic analysis of the upwind scheme for multi-dimensional transport equations. We associate a Markov chain with the numerical scheme and then obtain a backward representation formula of Kolmogorov type for the numerical solution. We then understand that the error induced by the scheme is governed by the fluctuations of the Markov chain around the characteristics of the flow. We show, in various situations, that the fluctuations are of diffusive type. As a by-product, we prove that the scheme is of order 1/2 for an initial datum in BV and of order 1/2-a, for all a>0, for a Lipschitz continuous initial datum. Our analysis provides a new interpretation of the numerical diffusion phenomenon.
Published: 2010
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26. A cell-centered Arbitrary Lagrangian Eulerian (ALE) method for multi-material compressible flows

Author: Pierre-Henri Maire, Cécile Dobrzynski, Aaron Diaz, Maya De Buhan, Gilles Kluth, Frédéric Lagoutière, Centre d'Etudes Lasers Intenses et Applications (CELIA), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-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), Department of Mathematics and Computer Science [Santa Clara], Santa Clara University, Institut de Mathématiques de Bordeaux (IMB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Algorithms and high performance computing for grand challenge applications (SCALAPPLIX), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), Département des Sciences de la Simulation et de l'Information (DSSI), DAM Île-de-France (DAM/DIF), Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Inria Bordeaux - Sud-Ouest, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Sciences et Technologies - Bordeaux 1-Université Bordeaux Segalen - Bordeaux 2
Subjects: Finite volume method, Discretization, Multi material, 010103 numerical & computational mathematics, Gas dynamics, Arbitrary lagrangian eulerian, 01 natural sciences, Compressible flow, 010101 applied mathematics, Classical mechanics, Scheme (mathematics), Compressibility, Applied mathematics, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: International audience; We present an original and accurate unstructured cell-centered ALE algorithm devoted to the simulation of two-dimensional multi-material compressible fluid flows. Gas dynamics equations are discretized with an unstructured finite volume scheme.
Published: 2008
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27. Lagrangian method enhanced with edge swapping for the free fall and contact problem

Author: Étienne Bernard, Stéphane Del Pino, Erwan Deriaz, Bruno Després, Katerina Jurkova, and Frédéric Lagoutière
Subjects: Physics, symbols.namesake, Mathematical analysis, Calculus, symbols, Edge (geometry), Deformation (meteorology), Lagrangian
Abstract: We discuss various algorithms that one can use in conjunction with Lagrangian methods for the treatment of important deformation of the mesh. We develop a new contact algorithm. The free fall of a droplet of fluid in a light gas serves to illustrate the potentialities of these techniques. Resume. Nousun probleme modele de chute libre d'une goutte de fluide dans de l'air ` l'aide de methodes numeriques lagrangiennes. Nous developpons etun nouvel algorithme de contact qui permet (numeriquement) ` la goutte de toucher le sol.
Published: 2008
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28. 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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29. Sharpening Methods for Finite Volume Schemes

Author: Samuel Kokh, Frédéric Lagoutière, and Bruno Després
Subjects: Finite volume method, Interface (Java), Advection, Computer science, Structure (category theory), 010103 numerical & computational mathematics, Sharpening, 01 natural sciences, 010101 applied mathematics, Nonlinear system, Simple (abstract algebra), 0101 mathematics, High order, Algorithm
Abstract: We review sharpening methods for finite volume schemes, with an emphasis on the basic structure of sharpening methods. It covers high order methods and non linear techniques for linear advection, Glimm's method, anti-diffusion techniques, the interaction of these techniques with the PDE structures. Additional approaches like level sets, interface reconstruction and Vofire are also discussed. We also present the algorithmic structure of the downwind method for a simple two components problem.
Published: 2016
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30. Sharpening methods for finite volume schemes

Author: Bruno Després, Samuel KOKH, Frédéric Lagoutière, 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), Maison de la Simulation (MDLS), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Institut National de Recherche en Informatique et en Automatique (Inria)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LMO), and Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Finite Volume schemes, Mathematics - Analysis of PDEs, interface tracking, FOS: Mathematics, anti-diffusion, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), Sharpening methods, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: International audience; We review sharpening methods for finite volume schemes, with an emphasis on the basic structure of sharpening methods. It covers high order methods and non linear techniques for linear advection, Glimm's method, anti-diffusion techniques, the interaction of these techniques with the PDE structures. Additional approaches like level sets, interface reconstruction and Vofire are also discussed. We also present the algorithmic structure of the downwind method for a simple two components problem.
Published: 2016
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31. LARGE TIME BEHAVIOR OF NUMERICAL SOLUTIONS OF SCALAR CONSERVATION LAWS

Author: Frédéric Lagoutière
Subjects: Conservation law, General Mathematics, Scheme (mathematics), Mathematical analysis, Convergence (routing), Scalar (physics), Analysis, Mathematics
Abstract: We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.
Published: 2006
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32. The Filippov characteristic flow for the aggregation equation with mildly singular potentials

Author: José A. Carrillo, Nicolas Vauchelet, Frédéric Lagoutière, François James, Department of Mathematics [Imperial College London], Imperial College London, Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), 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), 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 de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), ANR-13-BS01-0004,KIBORD,Modèles cinétiques en biologie et domaines connexes(2013), Université d'Orléans (UO)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), The Royal Society, and Engineering & Physical Science Research Council (E
Subjects: 35B40, 35D30, 35L60, 35Q92, 49K20, Dynamical systems theory, General Mathematics, KINETIC-MODELS, nonlocal conservation equations, 01 natural sciences, Convexity, 0101 Pure Mathematics, aggregation equation, gradient flow, Mathematics - Analysis of PDEs, DUALITY, 0102 Applied Mathematics, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Contraction (operator theory), Mathematics, GRANULAR FLOWS, Science & Technology, Applied Mathematics, 010102 general mathematics, Mathematical analysis, finite volume schemes, Filippov's flow, ASYMPTOTICS, CHEMOTAXIS, Pushforward measure, measure-valued solutions, EXISTENCE, 010101 applied mathematics, Metric space, UNIQUENESS, BLOWUP SOLUTIONS, Continuity equation, Physical Sciences, PATTERNS, Balanced flow, SYSTEM, Analysis, Analysis of PDEs (math.AP)
Abstract: Existence and uniqueness of global in time measure solution for the multidimensional aggregation equation is analyzed. Such a system can be written as a continuity equation with a velocity field computed through a self-consistent interaction potential. In Carrillo et al. (Duke Math J (2011)), a well-posedness theory based on the geometric approach of gradient flows in measure metric spaces has been developed for mildly singular potentials at the origin under the basic assumption of being lambda-convex. We propose here an alternative method using classical tools from PDEs. We show the existence of a characteristic flow based on Filippov's theory of discontinuous dynamical systems such that the weak measure solution is the pushforward measure with this flow. Uniqueness is obtained thanks to a contraction argument in transport distances using the lambda-convexity of the potential. Moreover, we show the equivalence of this solution with the gradient flow solution. Finally, we show the convergence of a numerical scheme for general measure solutions in this framework allowing for the simulation of solutions for initial smooth densities after their first blow-up time in Lp-norms., 33 pages
Published: 2014
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33. Generalized Harten Formalism and Longitudinal Variation Diminishing schemes for Linear Advection on Arbitrary Grids

Author: Frédéric Lagoutière and Bruno Després
Subjects: Perron–Frobenius theorem, Numerical Analysis, Finite volume method, Weak convergence, Applied Mathematics, Mathematical analysis, Computational Mathematics, Nonlinear system, Corollary, Modeling and Simulation, Bounded variation, Linear approximation, Analysis, Eigenvalues and eigenvectors, Mathematics
Abstract: We study a family of non linear schemes for the numerical solution of linear advection on arbitrary grids in several space dimension. A proof of weak convergence of the family of schemes is given, based on a new Longitudinal Variation Diminishing (LVD) estimate. This estimate is a multidimensional equivalent to the well-known TVD estimate in one dimension. The proof uses a corollary of the Perron-Frobenius theorem applied to a generalized Harten formalism.
Published: 2001
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34. A longitudinal variation diminishing estimate for linear advection on arbitrary grids

Author: Frédéric Lagoutière and Bruno Després
Subjects: Perron–Frobenius theorem, Advection, Mathematical analysis, Linear algebra, Stochastic matrix, Existence theorem, Upwind scheme, General Medicine, Grid, Variation (astronomy), Mathematics
Abstract: We prove a new longitudinal variation diminishing (LVD) estimate for linear advection on arbitrary grids. This estimate seems to be a multidimensional equivalent of the well known 1D TVD estimate.
Published: 2001
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35. [Untitled]

Author: Frédéric Lagoutière and Bruno Després
Subjects: Numerical Analysis, Discretization, Applied Mathematics, Mathematical analysis, General Engineering, Geometry, Euler system, Compressible flow, Theoretical Computer Science, Euler equations, Computational Mathematics, symbols.namesake, Discontinuity (linguistics), Computational Theory and Mathematics, symbols, Piecewise, Initial value problem, Uniform boundedness, Software, Mathematics
Abstract: We present a non-diffusive and contact discontinuity capturing scheme for linear advection and compressible Euler system. In the case of advection, this scheme is equivalent to the Ultra-Bee limiter of l24r, l29r. We prove for the Ultra-Bee scheme a property of exact advection for a large set of piecewise constant functions. We prove that the numerical error is uniformly bounded in time for such prepared (i.e., piecewise constant) initial data, and state a conjecture of non-diffusion at infinite time, based on some local over-compressivity of the scheme, for general initial data. We generalize the scheme to compressible gas dynamics and present some numerical results.
Published: 2001
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36. Un schéma non linéaire anti-dissipatif pour l'équation d'advection linéaire

Author: Frédéric Lagoutière and Bruno Després
Subjects: Pure mathematics, General Medicine, Mathematics
Abstract: Resume Nous presentons un schema anti-dissipatif obtenu par une methode de type ≪ limiteur ≫ non classique. Nous rappelons les proprietes VTD et de stabilite L ∞ pour ce schema. Nous donnons un theoreme d'adveetion exacte pour une classe de fonctions dense dans L 1 . Une conjecture quant au comportement des solutions numeriques en temps long est proposee.
Published: 1999
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37. 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

38. An Arbitrary Space-Time High-Order Finite Volume Scheme for Gas Dynamics Equations in Curvilinear Coordinates on Polar Meshes

Author: Stéphane Jaouen, Bertrand Meltz, and Frédéric Lagoutière
Subjects: Curvilinear coordinates, Classical mechanics, Generalized coordinates, Orthogonal coordinates, Parabolic cylindrical coordinates, Log-polar coordinates, Action-angle coordinates, Parabolic coordinates, Bipolar coordinates, Mathematics
Abstract: We are interested in the study of numerical schemes for the resolution of gas dynamics equations which preserve symmetric (or axisymmetric) flows. A simple way to achieve this is to derive a numerical scheme whose mesh and coordinates system are aligned with the flow. Typically, for the simulation of cylindrical implosions of gas, the cylindrical coordinate system and a polar mesh are well-suited. But such coordinates systems introduce geometrical singularities as well as geometrical source terms. In this paper, we investigate an arbitrary high-order space-time Finite Volume (FV) scheme in cylindrical coordinates. Test-cases with and without polar symmetries are studied in order to confirm the order of the scheme as well as its robustness.
Published: 2014
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39. 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

40. A second order anti-diffusive Lagrange-remap scheme for two-component flows

Author: Gloria Faccanoni, Frédéric Lagoutière, Benjamin Boutin, Filipa Caetano, Laurent Navoret, Marie Billaud Friess, Samuel Kokh, Centre d'Etudes Lasers Intenses et Applications ( CELIA ), Université de Bordeaux ( UB ) -Commissariat à l'énergie atomique et aux énergies alternatives ( CEA ) -Centre National de la Recherche Scientifique ( CNRS ), 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 des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Université Paris-Sud - Paris 11 ( UP11 ), Institut de Mathématiques de Toulon - EA 2134 ( IMATH ), Université de Toulon ( UTLN ), Laboratoire d'Etudes Thermiques des Réacteurs ( LETR ), Département de Modélisation des Systèmes et Structures ( DM2S ), Commissariat à l'énergie atomique et aux énergies alternatives ( CEA ) -Université Paris-Saclay-Commissariat à l'énergie atomique et aux énergies alternatives ( CEA ) -Université Paris-Saclay, Laboratoire de Recherche en Informatique ( LRI ), Université Paris-Sud - Paris 11 ( UP11 ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -CentraleSupélec-Centre National de la Recherche Scientifique ( CNRS ), Mathématiques Appliquées à Paris 5 ( MAP5 ), Institut National de la Recherche Agronomique ( INRA ) -Université Paris Descartes - Paris 5 ( UPD5 ) -Centre National de la Recherche Scientifique ( CNRS ), Centre d'Etudes Lasers Intenses et Applications (CELIA), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Centre National de la Recherche Scientifique (CNRS), 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), Université Paris-Sud - Paris 11 (UP11), Institut de Mathématiques de Toulon - EA 2134 (IMATH), Université de Toulon (UTLN), Laboratoire d'Etudes Thermiques des Réacteurs (LETR), Département de Modélisation des Systèmes et Structures (DM2S), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay-CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay, Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées à Paris 5 (MAP5), Institut National de la Recherche Agronomique (INRA)-Université Paris Descartes - Paris 5 (UPD5)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST-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 normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), CEA-Direction de l'Energie Nucléaire (CEA-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 de l'Energie Nucléaire (CEA-DEN), 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: T57-57.97, Mathematical optimization, Applied mathematics. Quantitative methods, Spacetime, Computer science, Compressible gas dynamics, 010103 numerical & computational mathematics, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Euler system, First order, 01 natural sciences, 010101 applied mathematics, Formalism (philosophy of mathematics), 65M06, 76M20, 76N15, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], QA1-939, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, 0101 mathematics, Mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We build a non-dissipative second order algorithm for the approximate resolution of the one-dimensional Euler system of compressible gas dynamics with two components. The considered model was proposed in [1]. The algorithm is based on [8] which deals with a non-dissipative first order resolution in Lagrange-remap formalism. In the present paper we describe, in the same framework, an algorithm that is second order accurate in time and space, and that preserves sharp interfaces. Numerical results reported at the end of the paper are very encouraging, showing the interest of the second order accuracy for genuinely non-linear waves. Nous construisons un algorithme d’ordre deux et non dissipatif pour la résolution approchée des équations d’Euler de la dynamique des gaz compressibles à deux constituants en dimension un. Le modèle que nous considérons est celui à cinq équations proposé et analysé dans [1]. L’algorithme est basé sur [8] qui proposait une résolution approchée à l’ordre un et non dissipative au moyen d’un splitting de type Lagrange-projection. Dans le présent article, nous décrivons, dans le même formalisme, un algorithme d’ordre deux en temps et en espace, qui préserve des interfaces « parfaites » entre les constituants. Les résultats numériques rapportés à la fin de l’article sont très encourageants ; ils montrent clairement les avantages d’un schéma d’ordre deux pour les ondes vraiment non linéaires.
Published: 2011
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41. The Lifschitz-Slyozov equation with space-diffusion of monomers

Author: Frédéric Lagoutière, Léon Matar Tine, Thierry Goudon, SImulations and Modeling for PArticles and Fluids (SIMPAF), Laboratoire Paul Painlevé - UMR 8524 (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 d'Analyse Numérique et Informatique [Sénégal] (LANI), Université Gaston Berger de Saint-Louis Sénégal (UGB), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), and Laboratoire Paul Painlevé (LPP)
Subjects: Physics, Numerical Analysis, Class (set theory), 010102 general mathematics, Kinetic energy, Space (mathematics), 01 natural sciences, 010101 applied mathematics, Condensed Matter::Soft Condensed Matter, chemistry.chemical_compound, Monomer, chemistry, Modeling and Simulation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, Diffusion (business)
Abstract: The Lifschitz--Slyozov system describes the dynamics of mass exchanges between macro--particles and monomers in the theory of coarsening. We consider a variant of the classical model where monomers are subject to space diffusion. We establish the existence--uniqueness of solutions for a wide class of relevant data and kinetic coefficients. We also derive a numerical scheme to simulate the behavior of the solutions.
Published: 2010

42. Small solids in an inviscid fluid

Author: Takéo Takahashi, Nicolas Seguin, Frédéric Lagoutière, Boris Andreianov, Laboratoire de Mathématiques de Besançon (UMR 6623) (LMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC), Laboratoire de Recherche en Informatique (LRI), Université Paris-Sud - Paris 11 (UP11)-CentraleSupélec-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 Élie Cartan de Nancy (IECN), 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), Robust control of infinite dimensional systems and applications (CORIDA), 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), 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, Institut National de Recherche en Informatique et en Automatique (Inria), CORIDA, 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), 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), and 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)
Subjects: Statistics and Probability, Solid-fluid interaction, 01 natural sciences, symbols.namesake, Quadratic equation, 35F25, 35L80, 65M99, Inviscid flow, singular source term, well-balanced scheme, random-choice method, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, adapted entropy, Mathematics, Partial differential equation, Finite volume method, Applied Mathematics, 010102 general mathematics, Mathematical analysis, General Engineering, Riemann solver, Computer Science Applications, Burgers' equation, Burgers equation, 010101 applied mathematics, Drag, Ordinary differential equation, symbols
Abstract: We present in this paper several results concerning a simple model of interaction between an inviscid fluid, modeled by the Burgers equation, and a particle, assumed to be point-wise. It is composed by a first-order partial differential equation which involves a singular source term and by an ordinary differential equation. The coupling is ensured through a drag force that can be linear or quadratic. Though this model can be considered as a simple one, its mathematical analysis is involved. We put forward a notion of entropy solution to our model, define a Riemann solver and make first steps towards well-posedness results. The main goal is to construct easy-to-implement and yet reliable numerical approximation methods; we design several finite volume schemes, which are analyzed and tested.
Published: 2010
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43. A Sharp Interface and Fully Conservative Scheme for Computing Nonclassical Shocks

Author: Benjamin Boutin, Frédéric Lagoutière, Christophe Chalons, and Philippe G. LeFloch
Subjects: symbols.namesake, Conservation law, Riemann problem, Scalar (mathematics), Mathematical analysis, symbols, Sharp interface, Applied mathematics, Quantum Physics, Numerical tests, Kinetic function, Mathematics
Abstract: We present a sharp interface and fully conservative numerical strategy for computing nonclassical solutions of scalar conservation laws. The difficult point is to impose at the discrete level a prescribed kinetic relation along each nonclassical discontinuity. Our method is based on a relevant reconstruction technique operating on each cell which is expected to contain a nonclassical shock. To prove the validity of our approach, we state some important stability properties and numerical tests are proposed. The convergence is also illustrated numerically.
Published: 2008
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44. Stability of reconstruction schemes for scalar hyperbolic conservations laws

Author: Frédéric Lagoutière
Subjects: entropy schemes, Conservation law, Finite volume method, Applied Mathematics, General Mathematics, Scalar (mathematics), Hyperbolic function, Mathematical analysis, reconstruction schemes, 65M12, hyperbolic equations, Numerical approximation, numerical schemes, 35L65, Hyperbolic partial differential equation, Approximate solution, Mathematics
Abstract: We study the numerical approximation of scalar conservation laws in dimension 1 via general reconstruction schemes within the finite volume framework. We exhibit a new stability condition, derived from an analysis of the spatial convolutions of entropy solutions with characteristic functions of intervals. We then propose a criterion that ensures the existence of some numerical entropy fluxes. The consequence is the convergence of the approximate solution to the unique entropy solution of the considered equation.
Published: 2008

45. Convergent and conservative schemes for nonclassical solutions based on kinetic relations

Author: Frédéric Lagoutière, Philippe G. LeFloch, Benjamin Boutin, 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), Laboratoire d'Etudes Thermiques des Réacteurs (LETR), 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)), and Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université Paris-Saclay
Subjects: [PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Discretization, Stability (learning theory), FOS: Physical sciences, 010103 numerical & computational mathematics, 01 natural sciences, Mathematics - Analysis of PDEs, Consistency (statistics), Convergence (routing), FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Mathematics, Conservation law, Applied Mathematics, 35L65, 76L05, Finite difference, Fluid Dynamics (physics.flu-dyn), Physics - Fluid Dynamics, Numerical Analysis (math.NA), 16. Peace & justice, Shock (mechanics), 010101 applied mathematics, Test case, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract: We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concave-convex as well as convex-concave flux-functions., Comment: 31 pages
Published: 2008
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46. A Relaxation Method for the Coupling of Systems of Conservation Laws

Author: A. Ambroso, Nicolas Seguin, Pierre-Arnaud Raviart, Christophe Chalons, Edwige Godlewski, Frédéric Lagoutière, and Frédéric Coquel
Subjects: Conservation law, Work (thermodynamics), symbols.namesake, Riemann problem, Computer science, Interface (Java), Numerical analysis, symbols, Complex system, Context (language use), Statistical physics, Coupling (probability)
Abstract: The present work investigates the spatial coupling of different hyperbolic models, from the theoritical and numerical points of view. The problem of coupling often occurs in an industrial context. Consider a complex system, such as a nuclear reactor. Different phenomena can appear according to the region and thus, different models (and numerical methods) are used. The difficulty lies in letting the different models communicate, since they are not necessarily compatible. Hence we study the spatial coupling of two different hyperbolic systems of conservation laws through the interface {x = 0}
Published: 2008
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47. Extension of Interface Coupling to General Lagrangian Systems

Author: Pierre-Arnaud Raviart, Christophe Chalons, Edwige Godlewski, Frédéric Coquel, A. Ambroso, Nicolas Seguin, and Frédéric Lagoutière
Subjects: symbols.namesake, Lagrangian and Eulerian specification of the flow field, Riemann hypothesis, Formalism (philosophy of mathematics), Riemann problem, Computer science, Numerical analysis, symbols, Applied mathematics, Gas dynamics, Boundary value problem, Lagrangian
Abstract: We study the coupling of two gas dynamics systems in Lagrangian coordinates at the interface x = 0. The coupling condition was formalized in [9], [10] by requiring that two boundary value problems should be well-posed, and it yields as far as possible the continuity of the solution at the interface. In this work we prove that we may choose the variables we transmit and extend the theory to Lagrangian systems of different sizes. The coupling condition is expressed in terms of Riemann problems, which is well suited for the numerical methods we are interested in implementing. Moreover this formalism is well adapted to Lagrangian systems since the sign of the wave speeds is known, which enables us to solve the coupled Riemann problem.
Published: 2007
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48. Numerical transport of an arbitrary number of components

Author: Frédéric Lagoutière, Stéphane Jaouen, 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), DAM Île-de-France (DAM/DIF), Direction des Applications Militaires (DAM), Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), and Joly, Pascal
Subjects: Conservation law, Finite volume method, Mechanical Engineering, Mathematical analysis, Computational Mechanics, Finite difference method, General Physics and Astronomy, Upwind scheme, Eulerian path, 010103 numerical & computational mathematics, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Compressible flow, Computer Science Applications, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mechanics of Materials, symbols, 0101 mathematics, Hyperbolic partial differential equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: This paper deals with the numerical transport of an arbitrary number of materials having the same velocity. One difficulty is to derive numerical algorithms that are conservative for the mass of each component and that satisfy some inequality and equality constraints: each mass fraction has to stay in [0, 1] and the sum of all mass fractions should be 1. These constraints are satisfied by the classical upwind scheme (which is very dissipative) but not for most of nonlinear (high-order or anti-dissipative) schemes. Here we propose local conditions of inequality type for the finite volume fluxes of mass fractions to ensure the aforementioned constraints. More precisely, we give explicit stability intervals for each flux. This is done in the manner of Despres and Lagoutiere [B. Despres, F. Lagoutiere, Contact discontinuity capturing schemes for linear advection and compressible gas dynamics, J. Sci. Comput. 16 (4) (2002) 479–524] for hyperbolic systems, Despres and Lagoutiere [B. Despres, F. Lagoutiere, Numerical resolution of a two-component compressible fluid model with interfaces, Progress Comput. Fluid Dyn., to appear] for the transport of 2 components; see also Bouchut [F. Bouchut, An anti-diffusive entropy scheme for monotone scalar conservation laws, J. Sci. Comput. 21 (1) (2004) 1–30] for the same type of inequality constraints for nonlinear conservation laws. Comparisons on two dimensional test-cases with the Youngs’ interface reconstruction algorithm [D.L. Youngs, An interface tracking method for a 3D eulerian hydrodynamics code, Technical Report 44/92/35, AWRE, 1984] show that results are qualitatively comparable. The advantages of this approach are its simplicity, its low computational cost, and its flexibility since it can deal with interfaces as well as mixing zones.
Published: 2006

49. Liquid jet generation and break-up

Author: Céline Baranger, G. Baudin, L. Boudin, Bruno Després, Frédéric Lagoutière, E. Lapébie, and Takéo Takahashi
Subjects: 010101 applied mathematics, 010102 general mathematics, 0101 mathematics, 01 natural sciences
Published: 2005
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50. A non-dissipative entropic scheme for convex scalar equations via discontinuous cell-reconstruction

Author: Frédéric Lagoutière, 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), and Ruprecht, Liliane
Subjects: Partial differential equation, Finite volume method, Numerical analysis, Computation, Scalar equation, Mathematical analysis, Regular polygon, 010103 numerical & computational mathematics, General Medicine, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, 010101 applied mathematics, Dissipative system, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: We present a finite volume non-dissipative but entropic scheme for convex scalar equations based on a discontinuous reconstruction of the solution in each cell of the mesh. This discontinuous representation of the numerical solution in each cell is done satisfying the L ∞ -norm, Total Variation and entropy decreasing properties. This allows us to prove the convergence towards the unique entropy solution. Numerical computations are reported, showing the non-dissipative behavior of the algorithm. To cite this article: F. Lagoutiere, C. R. Acad. Sci. Paris, Ser. I 338 (2004).
Published: 2004

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