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28 results on '"Nicolas Seguin"'

1. Stability of stationary solutions of singular systems of balance laws

Author

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

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

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

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

Author

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

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

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

Published
2020
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3. Convergence of finite volume schemes for the coupling between the inviscid Burgers equation and a particle

Author

Nina Aguillon, Nicolas Seguin, Frédéric Lagoutière, 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), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)

Subjects
PDE-ODE coupling, 01 natural sciences, Non-conservative coupling, convergence of finite volume schemes, Control theory, Inviscid flow, Convergence (routing), FOS: Mathematics, Mathematics - Numerical Analysis, 0101 mathematics, Fluid-particle interaction, Mathematics, Coupling, Pointwise, Algebra and Number Theory, Finite volume method, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Ode, Numerical Analysis (math.NA), Burgers equation, Burgers' equation, 010101 applied mathematics, Computational Mathematics, 35R37, 65M12, 35L72, Particle, moving interface, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this paper, we prove the convergence of a class of finite volume schemes for the model of coupling between a Burgers fluid and a pointwise particle introduced in [LST08]. In this model, the particle is seen as a moving interface through which an interface condition is imposed, which links the velocity of the fluid on the left and on the right of the particle and the velocity of the particle (the three quantities are all not equal in general). The total impulsion of the system is conserved through time.The proposed schemes are consistent with a “large enough” part of the interface conditions. The proof of convergence is an extension of the one of [AS12] to the case where the particle moves under the influence of the fluid. It yields two main difficulties: first, we have to deal with time-dependent flux and interface condition, and second with the coupling between and ODE and a PDE.

Published
2016
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4. Dissipative formulation of initial boundary value problems for Friedrichs’ systems

Author

Bruno Després, Clément Mifsud, Nicolas Seguin, Sorbonne Université (SU), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical Analysis, Geophysics and Ecology (ANGE), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

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

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

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

Author

Jens Klotzky, Nicolas Seguin, and Christian Klingenberg

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

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

Published
2018
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6. A finite-volume scheme for a kidney nephron model

Author

Nicolas Seguin, Aurélie Edwards, Magali Tournus, Department of Chemical and Biological Engineering, Tufts University [Medford], Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Centre de Recherche des Cordeliers (CRC (UMR_S 872)), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Descartes - Paris 5 (UPD5)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Université Paris Descartes - Paris 5 (UPD5)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut National de la Santé et de la Recherche Médicale (INSERM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions ( LJLL ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Centre de Recherche des Cordeliers ( CRC (UMR_S 872) ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National de la Santé et de la Recherche Médicale ( INSERM ) -Centre National de la Recherche Scientifique ( CNRS ), and Tournus, Magali

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

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

Published
2012
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7. Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

Author

Edwige Godlewski, Nicolas Seguin, Clément Cancès, Frédéric Coquel, Hélène Mathis, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, LRC MANON (CEA/DM2S -- LJLL) NEEDS program (CNRS -- CEA -- AREVA -- EDF -- IRSN), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Conservation law, Spacetime, Scale (ratio), Applied Mathematics, General Mathematics, Mathematical analysis, 35L65, 35B45, 35B30, 35A35, Flux, 010103 numerical & computational mathematics, Function (mathematics), error estimate, 01 natural sciences, 010101 applied mathematics, Nonlinear system, thick coupling interface, Ordinary differential equation, model adaptation, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Hyperbolic partial differential equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Conservation laws
Abstract

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

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

Author

Clément Mifsud, Jean-François Babadjian, Nicolas Seguin, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)

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

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

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

Author

Hélène Mathis, Nicolas Seguin, Clément Cancès, Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Nuclear System and Scenarios federative project of the NEEDS program (CNRS -- CEA -- AREVA -- EDF -- IRSN), LRC Manon (Modélisation et approximation numérique orientées pour l'énergie nucléaire - CEA/DM2S-LJLL)., ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)

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

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

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

Author

Julien Vovelle, Nicolas Seguin, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

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

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

Published
2003
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11. Dynamic model adaptation for multiscale simulation of hyperbolic systems with relaxation

Author

Nicolas Seguin, Hélène Mathis, Clément Cancès, Edwige Godlewski, Laboratoire de Mathématiques Jean Leray (LMJL), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), Reliable numerical approximations of dissipative systems (RAPSODI ), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Numerical Analysis, Geophysics and Ecology (ANGE), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, LRC Manon (Modélisation et approximation numérique orientées pour l'énergie nucléaire - CEA/DM2S-LJLL)., Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)

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

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

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

Author

Jean-Marc Hérard, Nicolas Seguin, and Thierry Gallouët

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

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

Published
2002
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13. A simple derivation of BV bounds for inhomogeneous relaxation systems

Author

Nicolas Seguin, Magali Tournus, Benoît Perthame, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Modelling and Analysis for Medical and Biological Applications (MAMBA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Pierre et Marie Curie - Paris 6 (UPMC), Numerical Analysis, Geophysics and Ecology (ANGE), Department of Mathematics [Philadelphia], University of Pennsylvania, and University of Pennsylvania [Philadelphia]

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

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

Published
2014
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14. MODELLING COMPRESSIBLE MULTIPHASE FLOWS

Author

Nicolas Seguin, Thierry Gallouët, Jean-Marc Hérard, Philippe Helluy, Olivier Hurisse, Frédéric Coquel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), TOkamaks and NUmerical Simulations (TONUS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), EDF (EDF), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)-Inria Nancy - Grand Est

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

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

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

Author

Paola Goatin, Nicolas Seguin, Christophe Chalons, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université Paris Diderot - Paris 7 (UPD7), Optimization and control, numerical algorithms and integration of complex multidiscipline systems governed by PDE (OPALE), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (LJAD), Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA), Université Pierre et Marie Curie - Paris 6 (UPMC), European Project: 257661,EC:FP7:ERC,ERC-2010-StG_20091028,TRAM3(2010), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Alexandre Dieudonné (JAD), Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UCA)-Université Nice Sophia Antipolis (... - 2019) (UNS), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS)

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

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

Published
2013

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

Author

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

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

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

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

Author

Nicolas Seguin, Frédéric Coquel, Jean-Marc Hérard, Khaled Saleh, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), EDF (EDF), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

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

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

Published
2012
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18. Error estimate for Godunov approximation of locally constrained conservation laws

Author

Nicolas Seguin, Clément Cancès, Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Conservation law, Finite volume method, Differential equation, Applied Mathematics, Mathematical analysis, Order (ring theory), BV-estimate, 010103 numerical & computational mathematics, error estimate, 01 natural sciences, monotone finite volume scheme, Locally constrained scalar conservation laws, 010101 applied mathematics, Constraint (information theory), Computational Mathematics, Monotone polygon, Numerical approximation, Convergence (routing), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Mathematics
Abstract

International audience; We consider a model of traffic flow with unilateral constraint on the flux introduced by {\sc R. M. Colombo} and {\sc P. Goatin} (J. Differ. Equ. 234(2):654--675, 2007), for which the convergence of numerical approximation using monotone finite volume schemes has been performed by B. {\sc Andreianov} {\em et al.}~(Numer. Math. 115:609--645, 2010). We derive for this problem some new ${\rm BV}$-estimate, and make use of it to provide an error estimate for the Godunov approximation of the problem of order $h^{1/3}$ that is improved into the optimal order $h^{1/2}$ under a reasonable assumption. Numerical experiments are then provided to illustrate the optimality of the result.

Published
2012
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19. Finite volume schemes for locally constrained conservation laws

Author

Paola Goatin, Boris Andreianov, Nicolas Seguin, 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), Institut des Sciences de l'Ingénieur de Toulon et du Var (ISITV), Université de Toulon (UTLN), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Université de Bourgogne (UB)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Conservation law, MSC 35L65, 65M12, 76M12, 90B20, Finite volume method, Applied Mathematics, Numerical analysis, 010102 general mathematics, Mathematical analysis, [INFO.INFO-NA]Computer Science [cs]/Numerical Analysis [cs.NA], 01 natural sciences, Green wave, 010101 applied mathematics, Computational Mathematics, Monotone polygon, 0101 mathematics, Road traffic, Mathematics
Abstract

This paper is devoted to the numerical analysis of the road traffic model proposed by Colombo and Goatin (J. Differ. Equ. 234(2):654–675, 2007). The model involves a standard conservation law supplemented by a local unilateral constraint on the flux at the point x = 0 (modelling a road light, a toll gate, etc.). We first show that the problem can be interpreted in terms of the theory of conservation laws with discontinuous flux function, as developed by Adimurthi et al. (J. Hyperbolic Differ. Equ. 2(4):783–837, 2005) and Burger et al. (SIAM J. Numer. Anal. 47(3):1684–1712, 2009). We reformulate accordingly the notion of entropy solution introduced by Colombo and Goatin (J. Differ. Equ. 234(2):654–675, 2007), and extend the well-posedness results to the L ∞ framework. Then, starting from a general monotone finite volume scheme for the non-constrained conservation law, we produce a simple scheme for the constrained problem and show its convergence. The proof uses a new notion of entropy process solution. Numerical examples modelling a “green wave” are presented.

Published
2010
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20. 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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21. Coupling of general Lagrangian systems

Author

A. Ambroso, Edwige Godlewski, Pierre-Arnaud Raviart, Christophe Chalons, Frédéric Coquel, Frédéric Lagoutière, Nicolas Seguin, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), CEA-Direction des Energies (ex-Direction de l'Energie Nucléaire) (CEA-DES (ex-DEN)), Commissariat à l'énergie atomique et aux énergies alternatives (CEA), and CEA-Direction de l'Energie Nucléaire (CEA-DEN)

Subjects
Work (thermodynamics), [PHYS.NUCL]Physics [physics]/Nuclear Theory [nucl-th], Interface (Java), media_common.quotation_subject, Basic methods in fluid mechanics, 010103 numerical & computational mathematics, [PHYS.NEXP]Physics [physics]/Nuclear Experiment [nucl-ex], 01 natural sciences, symbols.namesake, Lagrangian and Eulerian specification of the flow field, 0101 mathematics, Mathematics, media_common, Numerical Analysis, Algebra and Number Theory, Variables, Applied Mathematics, Numerical analysis, Mathematical analysis, 010101 applied mathematics, Computational Mathematics, Coupling (physics), Riemann problem, Partial differential equations of hyperbolic type, symbols, Euler's formula
Abstract

International audience; This work is devoted to the coupling of two fluid models, such as two Euler systems in Lagrangian coordinates, at a fixed interface. We define coupling conditions which can be expressed in terms of continuity of some well chosen variables and then solve the coupled Riemann problem. In the present setting where the interface is charactersitic, a particular choice of dependent variables which are transmitted ensures the overall conservativity.

Published
2008
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22. Relaxation models of phase transition flows

Author

Philippe Helluy, Nicolas Seguin, Institut de Recherche Mathématique Avancée (IRMA), Université Louis Pasteur - Strasbourg I-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), and Helluy, Philippe

Subjects
Numerical Analysis, Phase transition, Finite volume method, Mathematical model, Applied Mathematics, Numerical analysis, Constrained optimization, Relaxation (iterative method), Thermodynamics, 010103 numerical & computational mathematics, 01 natural sciences, 010101 applied mathematics, Physics::Fluid Dynamics, Computational Mathematics, Modeling and Simulation, Convex optimization, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Entropy (arrow of time), Analysis, Mathematics
Abstract

In this work, we propose a general framework for the construction of pressure law for phase transition. These equations of state are particularly suitable for a use in a relaxation finite volume scheme. The approach is based on a constrained convex optimization problem on the mixture entropy. It is valid for both miscible and immiscible mixtures. We also propose a rough pressure law for modelling a super-critical fluid.

Published
2006

23. The Riemann problem for a simple model of phase transition

Author

Edwige Godlewski, Nicolas Seguin, Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Ruprecht, Liliane

Subjects
Physics, Phase transition, Van der Waals equation, Applied Mathematics, General Mathematics, 010103 numerical & computational mathematics, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 01 natural sciences, Isothermal process, 010101 applied mathematics, symbols.namesake, Riemann problem, Simple (abstract algebra), symbols, 0101 mathematics, Constant (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematical physics
Abstract

We consider in this paper a simple model of liquid-vapor phase transition. The PDE system corresponds to the isothermal $p$-system and the pressure law is defined for all positive density and involves a constant zone. Such a pressure law can occur for instance when considering the Van der Waals equation of state complemented by the Maxwell's law. The Riemann problem is globally solved for all initial states, as well in the pure phases as in the mixture zone.

Published
2006

24. Numerical simulations of wave breaking

Author

Anne Cecile Lesage, Jean-Paul Caltagirone, Nicolas Seguin, Déborah Drevard, Olivier Allain, Richard Marcer, Stephan T. Grilli, Alain Dervieux, Stéphane Vincent, Pierre Lubin, Philippe Fraunié, Philippe Helluy, Frederic Golay, 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
Numerical Analysis, Computer simulation, Mathematical model, Applied Mathematics, Numerical analysis, Multiphase flow, Breaking wave, 01 natural sciences, Compressible flow, 010305 fluids & plasmas, 010101 applied mathematics, Computational Mathematics, Incompressible flow, Modeling and Simulation, 0103 physical sciences, Compressibility, Statistical physics, 0101 mathematics, Analysis, Mathematics
Abstract

This paper is devoted to the numerical simulation of wave breaking. It presents the results of a numerical workshop that was held during the conference LOMA04. The objective is to compare several mathematical models (compressible or incompressible) and associated numerical methods to compute the flow field during a wave breaking over a reef. The methods will also be compared with experiments.

Published
2005

25. Numerical modeling of two-phase flows using the two-fluid two-pressure approach

Author

Ronald Remmerswaal, Nicolas Seguin, Jean-Marc Hérard, Laboratoire d'Analyse, Topologie, Probabilités (LATP), Université Paul Cézanne - Aix-Marseille 3-Université de Provence - Aix-Marseille 1-Centre National de la Recherche Scientifique (CNRS), EDF R&D (EDF R&D), EDF (EDF), Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Shock wave, Finite Volume, AMS: 76T10, 35F25, 35L67, 76M12, Classification of discontinuities, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, Maximum principle, 0103 physical sciences, 0101 mathematics, Mathematics, Finite volume method, Applied Mathematics, Mathematical analysis, Relaxation (iterative method), Godunov's scheme, non-conservative terms, 010101 applied mathematics, two-phase flow, Riemann problem, hyperbolic system, resonance, Modeling and Simulation, symbols, Two-phase flow, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The present paper is devoted to the computation of two-phase flows using the two-fluid approach. The overall model is hyperbolic and has no conservative form. No instantaneous local equilibrium between phases is assumed, which results in a two-velocity two-pressure model. Original closure laws for interfacial velocity and interfacial pressure are proposed. These closures allow to deal with discontinuous solutions such as shock waves and contact discontinuities without ambiguity with the definition of Rankine–Hugoniot jump relations. Each field of the convective system is investigated, providing maximum principle for the volume fraction and the positivity of densities and internal energies are ensured when focusing on the Riemann problem. Two-finite volume methods are presented, based on the Rusanov scheme and on an approximate Godunov scheme. Relaxation terms are taken into account using a fractional step method. Eventually, numerical tests illustrate the ability of both methods to compute two-phase flows.

Published
2004

26. Hybrid schemes to compute contact discontinuities in Euler equations with any EOS

Author

Nicolas Seguin, Jean-Marc Hérard, Thierry Gallouët, Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Simulation et Traitement de l'information pour l'Exploitation des systèmes de Production (EDF R&D STEP), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Equation of state, Strategy and Management, Computation, MathematicsofComputing_NUMERICALANALYSIS, 010103 numerical & computational mathematics, Classification of discontinuities, Computational fluid dynamics, 01 natural sciences, Mathematics::Numerical Analysis, symbols.namesake, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, Media Technology, Applied mathematics, General Materials Science, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [MATH]Mathematics [math], Mathematics, Physics::Computational Physics, Marketing, business.industry, Godunov's theorem, Godunov's scheme, Euler equations, 010101 applied mathematics, Discontinuity (linguistics), Classical mechanics, symbols, business
Abstract

International audience; We propose here some explicit hybrid schemes which enable accurate computation of Euler equations with arbitrary (analytic or tabulated) equation of state (EOS). The method is valid for the exact Godunov scheme and some approximate Godunov schemes.; On propose un schéma explicite hybride permettant d’effectuer des simulations précises des équations d’Euler pour un choix de loi thermodynamique quelconque, analytique ou tabulée. La méthode est définie pour les schémas de Godunov exact ou approchés.

Published
2002
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27. RELAXATION OF FLUID SYSTEMS

Author

Nicolas Seguin, Edwige Godlewski, Frédéric Coquel, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Jacques-Louis Lions (LJLL), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), and LRC Manon (Laboratoire de recherche conventionné -- CEA/DM2S-LJLL -- Modélisation et approximation numérique orientées pour l'énergie nucléaire)

Subjects
010103 numerical & computational mathematics, 01 natural sciences, Godunov-type scheme, symbols.namesake, finite volume scheme, Hyperbolic system, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, fluid model, 0101 mathematics, Mathematics, Applied Mathematics, Numerical analysis, Degenerate energy levels, Mathematical analysis, Godunov's scheme, relaxation approximation, Solver, 16. Peace & justice, 010101 applied mathematics, Riemann problem, Lagrangian relaxation, Modeling and Simulation, symbols, Euler's formula, Relaxation (approximation), AMSC: 35L60, 65M08, 76M12, 35B25, 35Q35, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We propose a relaxation framework for general fluid models which can be understood as a natural ex- tension of the Suliciu approach in the Euler setting. In particular, the relaxation system may be totally degenerate. Several stability properties are proved. The relaxation procedure is shown to be efficient in the numerical approximation of the entropy weak solutions of the original PDEs. The numerical method is particularly simple in the case of a fully degenerate relaxation system for which the solution of the Riemann problem is explicit. Indeed, the Godunov solver for the homogeneous relaxation system results in an HLLC-type solver for the equilibirum model. Discrete entropy inequalities are established under a natural Gibbs principle.

Published
2012
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28. A simple 1D model of inviscid fluid–solid interaction

Author

Frédéric Lagoutière, Takéo Takahashi, Nicolas Seguin, 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), 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, and 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

Subjects
Shock wave, 01 natural sciences, Riemann problem, symbols.namesake, Inviscid flow, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Fluid-particle interaction, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Fluid–particle interaction, Interaction model, 16. Peace & justice, Non-conservative product, Burgers' equation, Burgers equation, 010101 applied mathematics, Classical mechanics, Flow velocity, Ordinary differential equation, symbols, Analysis, 76N20, 35B40, 35Q53
Abstract

International audience; We analyze a one-dimensional fluid-particle interaction model, composed by the Burgers equation for the fluid velocity and an ordinary differential equation which governs the particle movement. The coupling is achieved through a friction term. One of the novelties is to consider entropy weak solutions involving shock waves. The difficulty is the interaction between these shock waves and the particle. We prove that the Riemann problem with arbitrary data always admits a solution, which is explicitly constructed. Besides, two asymptotic behaviors are described: the long-time behavior and the behavior for large friction coefficients.

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