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23 results on '"Bruno Dubroca"'

1. A kinetic approach of the bi-temperature Euler model

Author

Stéphane Brull, Bruno Dubroca, Corentin Prigent, Institut de Mathématiques de Bordeaux (IMB), 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), Laboratoire des Composites Thermostructuraux (LCTS), Centre National de la Recherche Scientifique (CNRS)-Snecma-SAFRAN group-Université de Bordeaux (UB)-Institut de Chimie du CNRS (INC)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA), Centre d'Etudes Lasers Intenses et Applications (CELIA), 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 (UB)-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Centre National de la Recherche Scientifique (CNRS), and Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Institut de Chimie du CNRS (INC)-Snecma-SAFRAN group-Centre National de la Recherche Scientifique (CNRS)

Subjects
Kinetic scheme, 010103 numerical & computational mathematics, 01 natural sciences, Stability (probability), symbols.namesake, asympotic preserving scheme, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Scaling, Mathematics, Numerical Analysis, plasma physics, Numerical analysis, Mathematical analysis, Order (ring theory), [CHIM.MATE]Chemical Sciences/Material chemistry, nonconservative hyperbolic system, Euler equations, 010101 applied mathematics, Modeling and Simulation, symbols, Euler's formula, Bgk model, kinetic scheme, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We are interested in the numerical approximation of the bi-temperature Euler equations, which is a non conservative hyperbolic system introduced in [ 4 ]. We consider a conservative underlying kinetic model, the Vlasov-BGK-Poisson system. We perform a scaling on this system in order to obtain its hydrodynamic limit. We present a deterministic numerical method to approximate this kinetic system. The method is shown to be Asymptotic-Preserving in the hydrodynamic limit, which means that any stability condition of the method is independant of any parameter \begin{document}$ \varepsilon $\end{document} , with \begin{document}$ \varepsilon \rightarrow 0 $\end{document} . We prove that the method is, under appropriate choices, consistant with the solution for bi-temperature Euler. Finally, our method is compared to methods for the fluid model (HLL, Suliciu).

Published
2020

2. The $M_1$ Angular Moments Model in a Velocity-Adaptive Frame for Rarefied Gas Dynamics Applications

Author

Bruno Dubroca, Denise Aregba, Sébastien Guisset, and Stéphane Brull

Subjects
Physics, Zero mean, Work (thermodynamics), Conservation law, Galilean invariance, Ecological Modeling, Frame (networking), General Physics and Astronomy, Order (ring theory), 010103 numerical & computational mathematics, General Chemistry, Gas dynamics, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, Classical mechanics, Modeling and Simulation, 0101 mathematics
Abstract

In the present work the $M_1$ angular moments model in a velocity-adaptive frame is presented for rarefied gas dynamics applications. First of all, the derivation of the angular $M_1$ moments model in the particle mean velocity frame is introduced. The choice of the mean velocity framework in order to enforce the Galilean invariance property of the model is highlighted. In addition, it is shown that the model rewritten in terms of the entropic variables is Friedrichs-symmetric. Also, the derivation of the associated conservation laws and the zero mean velocity condition are detailed. Second, a suitable numerical scheme, preserving the realizable requirement of the numerical solution for the angular $M_1$ moments model in the mean velocity frame, is proposed. Third, some numerical results obtained considering several test cases in different collisional regimes are displayed.

Published
2017

3. Modelling and entropy satisfying relaxation scheme for the nonconservative bitemperature Euler system with transverse magnetic field

Author

Bruno Dubroca, Stéphane Brull, Xavier Lhébrard, lhebrard, xavier, Institut de Mathématiques de Bordeaux (IMB), 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), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), 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 Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), 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 ), Laboratoire d'Analyse et de Mathématiques Appliquées ( LAMA ), Université Paris-Est Marne-la-Vallée ( UPEM ) -Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 ( UPEC UP12 ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bordeaux ( IMB ), and 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 )

Subjects
General Computer Science, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, Entropy (classical thermodynamics), hydrodynamic limit, Relaxation system, 0103 physical sciences, discrete entropy minimum principle, 0101 mathematics, BGK models, Physics, Finite volume method, relaxation method, Mathematical analysis, General Engineering, Relaxation (iterative method), [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], Euler system, Solver, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], non-conservative hyperbolic system, discrete entropy inequalities, 010101 applied mathematics, Maxwell's equations, Transverse magnetic field, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

The present paper concerns the study of the nonconservative bitem-perature Euler system with transverse magnetic field. We firstly introduce an underlying conservative kinetic model coupled to Maxwell equations. The nonconservative bitemperature Euler system with transverse magnetic field is then established from this kinetic model by hydrodynamic limit. Next we present the derivation of a finite volume method to approximate weak solutions. It is obtained by solving a relaxation system of Suliciu type, and is similar to HLLC type solvers. The solver is shown in particular to preserve positivity of density and internal energies. Moreover we use a local minimum entropy principle to prove discrete entropy inequalities, ensuring the robustness of the scheme.

Published
2019

4. An Approximation of the $$M_2$$ M 2 Closure: Application to Radiotherapy Dose Simulation

Author

Bruno Dubroca, T. Pichard, Stéphane Brull, Martin Frank, and G. W. Alldredge

Subjects
Numerical Analysis, Computer simulation, business.industry, Applied Mathematics, Computation, Mathematical analysis, General Engineering, 010103 numerical & computational mathematics, Solver, 01 natural sciences, Theoretical Computer Science, 010101 applied mathematics, Computational Mathematics, Test case, Software, Computational Theory and Mathematics, Closure (computer programming), Realizability, Entropy (information theory), 0101 mathematics, business, Mathematics
Abstract

Particle transport in radiation therapy can be modelled by a kinetic equation which must be solved numerically. Unfortunately, the numerical solution of such equations is generally too expensive for applications in medical centers. Moment methods provide a hierarchy of models used to reduce the numerical cost of these simulations while preserving basic properties of the solutions. Moment models require a closure because they have more unknowns than equations. The entropy-based closure is based on the physical description of the particle interactions and provides desirable properties. However, computing this closure is expensive. We propose an approximation of the closure for the first two models in the hierarchy, the $$M_1$$M1 and $$M_2$$M2 models valid in one, two or three dimensions of space. Compared to other approximate closures, our method works in multiple dimensions. We obtain the approximation by a careful study of the domain of realizability and by invariance properties of the entropy minimizer. The $$M_2$$M2 model is shown to provide significantly better accuracy than the $$M_1$$M1 model for the numerical simulation of a dose computation in radiotherapy. We propose a numerical solver using those approximated closures. Numerical experiments in dose computation test cases show that the new method is more efficient compared to numerical solution of the minimum entropy problem using standard software tools.

Published
2016

5. TheM2Model for Dose Simulation in Radiation Therapy

Author

Bruno Dubroca, Graham W. Alldredge, T. Pichard, Martin Frank, and Stéphane Brull

Subjects
Physics, Angular momentum, Photon, Kinetic model, Applied Mathematics, Computation, Order up to, medicine.medical_treatment, General Physics and Astronomy, Transportation, Statistical and Nonlinear Physics, 010103 numerical & computational mathematics, Electron, 01 natural sciences, Computational physics, 010101 applied mathematics, Radiation therapy, medicine, 0101 mathematics, Atomic physics, Entropy (energy dispersal), Mathematical Physics
Abstract

The transport of photons and electrons is studied in the field of radiotherapy to compute the dose, that is, the quantity of energy transferred to the medium by a beam of particles at each position. A kinetic model is proposed, and to decrease the computation times, it is reduced through a moment extraction. Entropy-based angular moment models of order up to two (M1 and M2 models) are shown to provide accurate results compared to a reference code with much lower computational costs.

Published
2016

6. Classical transport theory for the collisional electronic M1 model

Author

Stéphane Brull, V. T. Tikhonchuk, Emmanuel d'Humières, Sébastien Guisset, and Bruno Dubroca

Subjects
Statistics and Probability, Physics, Work (thermodynamics), Statistical and Nonlinear Physics, Electron, Plasma, Collision, Kinetic energy, 01 natural sciences, Electron transport chain, 010305 fluids & plasmas, 010101 applied mathematics, Physics::Plasma Physics, Quantum electrodynamics, Quantum mechanics, 0103 physical sciences, Moment (physics), Limit (mathematics), 0101 mathematics
Abstract

The electronic M 1 model is widely used for electron transport studies in a hot collisional plasma. However, the moment extraction of the electron–electron collision operator from the kinetic collision operator, for this angular moments model, is challenging and some approximations are required. In this work a characterisation of the electron–electron and electron–ion collision operators is given and the electron plasma transport coefficients are derived. It is shown that in the high Z limit the electronic M 1 model and the Fokker–Planck–Landau equation coincide in the case of near equilibrium. Also, in general, the electron–electron collision operator proposed for the electronic M 1 model recovers accurate electron transport plasma coefficients.

Published
2016

7. Asymptotic-Preserving Scheme for the M1-Maxwell System in the Quasi-Neutral Regime

Author

Bruno Dubroca, I. Potapenko, S. Karpov, Stéphane Brull, Sébastien Guisset, and Emmanuel d'Humières

Subjects
Physics, Work (thermodynamics), Angular momentum, Physics and Astronomy (miscellaneous), Spacetime, Stiffness, 010103 numerical & computational mathematics, Kinetic energy, 01 natural sciences, 010101 applied mathematics, symbols.namesake, Maxwell's equations, Scheme (mathematics), medicine, symbols, Limit (mathematics), Statistical physics, 0101 mathematics, medicine.symptom
Abstract

This work deals with the numerical resolution of the M1-Maxwell system in the quasi-neutral regime. In this regime the stiffness of the stability constraints of classical schemes causes huge calculation times. That is why we introduce a new stable numerical scheme consistent with the transitional and limit models. Such schemes are called Asymptotic-Preserving (AP) schemes in literature. This new scheme is able to handle the quasi-neutrality limit regime without any restrictions on time and space steps. This approach can be easily applied to angular moment models by using a moments extraction. Finally, two physically relevant numerical test cases are presented for the Asymptotic-Preserving scheme in different regimes. The first one corresponds to a regime where electromagnetic effects are predominant. The second one on the contrary shows the efficiency of the Asymptotic-Preserving scheme in the quasi-neutral regime. In the latter case the illustrative simulations are compared with kinetic and hydrodynamic numerical results.

Published
2016

8. Modelling and numerical approximation for the nonconservative bitemperature Euler model

Author

Bruno Dubroca, Stéphane Brull, Jérôme Breil, Denise Aregba-Driollet, Elise Estibals, 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), 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), Équipe Calcul scientifique et Modélisation, 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), Control, Analysis and Simulations for TOkamak Research (CASTOR), 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 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)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), 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), and 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)

Subjects
Relaxation method, Kinetic energy, 01 natural sciences, 010305 fluids & plasmas, symbols.namesake, hydrodynamic limit, 0103 physical sciences, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, kinetic schemes, BGK models, Mathematics, Numerical Analysis, Applied Mathematics, Numerical analysis, entropy dissipation, Relaxation (iterative method), Mathematics Subject Classification: 65M08 / 35L60 / 35L65 / 82D10 / 76X05, Dissipation, Euler system, nonconservative hyperbolic system, 010101 applied mathematics, Computational Mathematics, Classical mechanics, Modeling and Simulation, Euler's formula, symbols, Dissipative system, Poisson's equation, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; This paper is devoted to the study of the nonconservative bitemperature Euler system. We firstly introduce an underlying two species kinetic model coupled with the Poisson equation. The bitemperature Euler system is then established from this kinetic model according to an hydrodynamic limit. A dissipative entropy is proved to exist and a solution is defined to be admissible if it satisfies the related dissipation property. Next, four different numerical methods are presented. Firstly, the kinetic model gives rise to kinetic schemes for the fluid system. The second approach belongs to the family of the discrete BGK schemes introduced by Aregba-Driollet and Natalini. Finally, a quasi-linear relaxation approach and a Lagrange-remap scheme are considered. 1991 Mathematics Subject Classification. 65M08, 35L60, 35L65. Secondary: 82D10, 76X05. The dates will be set by the publisher.

Published
2018

9. On the Transverse Diffusion of Beams of Photons in Radiation Therapy

Author

Stéphane Brull, T. Pichard, Martin Frank, Bruno Dubroca, 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 d'Etudes Lasers Intenses et Applications (CELIA), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), Institut de Mathématiques de Bordeaux (IMB), 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), Rheinisch-Westfälische Technische Hochschule Aachen (RWTH), Pichard, Teddy, Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-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), and Rheinisch-Westfälische Technische Hochschule Aachen University (RWTH)

Subjects
Physics, [PHYS.PHYS.PHYS-MED-PH] Physics [physics]/Physics [physics]/Medical Physics [physics.med-ph], Photon, Numerical analysis, Significant difference, 010103 numerical & computational mathematics, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Numerical diffusion, 01 natural sciences, External radiotherapy, implicit scheme, Computational physics, 010101 applied mathematics, Transverse plane, Classical mechanics, [PHYS.PHYS.PHYS-MED-PH]Physics [physics]/Physics [physics]/Medical Physics [physics.med-ph], M1 model, relaxation scheme, 0101 mathematics, Photon beam, Photon diffusion, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Typical external radiotherapy treatments consist in emitting beams of energetic photons targeting the tumor cells. Those photons are transported through the medium and interact with it. Such interactions affect the motion of the photons but they are typically weakly deflected which is not well modeled by standard numerical methods. The present work deals with the transport of photons in water. The motion of those particles is modeled by an entropy-based moment model, i.e. the M 1 model. The main difficulty when constructing numerical approaches for photon beam modelling emerges from the significant difference of magnitude between the diffusion effects in the forward and transverse directions. A numerical method for the M 1 equations is proposed with a special focus on the numerical diffusion effects.

Published
2018

10. Relaxation schemes for the M 1 model with space-dependent flux: application to radiotherapy dose calculation

Author

Stéphane Brull, Bruno Dubroca, T. Pichard, Martin Frank, Denise Aregba-Driollet, Centre d'Etudes Lasers Intenses et Applications (CELIA), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), Institut de Mathématiques de Bordeaux (IMB), 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 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), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-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), 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), and Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Moments models, Physics and Astronomy (miscellaneous), Radiotherapy, Computer science, Computer Science::Information Retrieval, Numerical analysis, Scalar (mathematics), 010103 numerical & computational mathematics, relaxation models, System of linear equations, 01 natural sciences, Stencil, AMS subject classifications: 82C40, 35L65, 65M25, 010101 applied mathematics, Method of characteristics, Realizability, Radiotherapy dose, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 0101 mathematics, method of characteristics, Difference quotient, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Because of stability constraints, most numerical schemes applied to hyperbolic systems of equations turn out to be costly when the flux term is multiplied by some very large scalar. This problem emerges with the M1 system of equations in the field of radiotherapy when considering heterogeneous media with very disparate densities. Additionally, the flux term of the M1 system is non-linear, and in order for the model to be well-posed the numerical solution needs to fulfill conditions called realizability. In this paper, we propose a numerical method that overcomes the stability constraint and preserves the realizability property. For this purpose, we relax the M1 system to obtain a linear flux term. Then we extend the stencil of the difference quotient to obtain stability. The scheme is applied to a radiotherapy dose calculation example.

Published
2016

11. An entropy preserving relaxation scheme for ten-moments equations with source terms

Author

Bruno Dubroca, Afeintou Sangam, Christophe Berthon, 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), 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 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), Control, Analysis and Simulations for TOkamak Research (CASTOR), 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), Laboratoire Jean Alexandre Dieudonné (LJAD), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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), 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), 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), and Laboratoire Jean Alexandre Dieudonné (JAD)

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], General Mathematics, 010103 numerical & computational mathematics, 01 natural sciences, [PHYS.PHYS.PHYS-PLASM-PH]Physics [physics]/Physics [physics]/Plasma Physics [physics.plasm-ph], Hyperbolic system, Godunov type schemes, [MATH]Mathematics [math], 0101 mathematics, Ten-Moments equations, Entropy rate, Mathematics, source terms, Finite volume method, Shannon's source coding theorem, Applied Mathematics, Principle of maximum entropy, 65M12, Mathematical analysis, Maximum entropy thermodynamics, discrete entropy minimum principle AMS subject classifications 65M60, discrete entropy inequalities, Quantum relative entropy, 010101 applied mathematics, Maximum entropy probability distribution, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Joint quantum entropy
Abstract

International audience; The present paper concerns the derivation of finite volume methods to approximate weak solutions of Ten-Moments equations with source terms. These equations model compressible anisotropic flows. A relaxation-type scheme is proposed to approximate such flows. Both robustness and stability conditions of the suggested finite volume methods are established. To prove discrete entropy inequalities, we derive a new strategy based on local minimum entropy principle and never use some approximate PDE's auxiliary model as usually recommended. Moreover, numerical simulations in 1D and in 2D illustrate our approach.

Published
2015

12. Reduced entropic model for studies of multidimensional nonlocal transport in high-energy-density plasmas

Author

Bruno Dubroca, M. Touati, Vladimir Tikhonchuk, Jean-Luc Feugeas, Sébastien Guisset, Ph. Nicolaï, D. Del Sorbo, and M. Olazabal-Loumé

Subjects
Physics, H-theorem, Electron, Condensed Matter Physics, Kinetic energy, 01 natural sciences, 7. Clean energy, Boltzmann equation, 010305 fluids & plasmas, Distribution function, Heat flux, 0103 physical sciences, Heat transfer, Fokker–Planck equation, Statistical physics, 010306 general physics
Abstract

Hydrodynamic simulations of high-energy-density plasmas require a detailed description of energy fluxes. For low and intermediate atomic number materials, the leading mechanism is the electron transport, which may be a nonlocal phenomenon requiring a kinetic modeling. In this paper, we present and test the results of a nonlocal model based on the first angular moments of a simplified Fokker-Planck equation. This multidimensional model is closed thanks to an entropic relation (the Boltzman H-theorem). It provides a better description of the electron distribution function, thus enabling studies of small scale kinetic effects within the hydrodynamic framework. Examples of instabilities of electron plasma and ion-acoustic waves, driven by the heat flux, are presented and compared with the classical formula.

Published
2015
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13. Partial moment entropy approximation to radiative heat transfer

Author

Bruno Dubroca, Martin Frank, Axel Klar, Departement of Mathematics, TU Kaiserslautern, University of Kaiserslautern, Institut de Mathématiques de Bordeaux (IMB), 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), and Publica

Subjects
Unit sphere, Numerical Analysis, Physics and Astronomy (miscellaneous), Applied Mathematics, Geometry, 010103 numerical & computational mathematics, Half-space, 01 natural sciences, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Entropy (classical thermodynamics), Moment closure, Thermal radiation, Modeling and Simulation, Heat transfer, Radiative transfer, SPHERES, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematical physics, Mathematics
Abstract

We extend the half moment entropy closure for the radiative heat transfer equations presented in Dubroca and Klar [B. Dubroca, A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys. 180 (2002) 584-596] and Turpault et al. [R. Turpault, M. Frank, B. Dubroca, A. Klar, Multigroup half space moment approximations to the radiative heat transfer equations, J. Comput. Phys. 198 (2004) 363-371] to multi-D. To that end, we consider a partial moment system with general partitions of the unit sphere closed by an entropy minimization principle. We give physical and mathematical reasons for this choice of model and study its properties. Several numerical examples in different physical regimes are presented.

Published
2006

14. An entropic scheme for an angular moment model for the classical Fokker-Planck-Landau equation of electrons

Author

Stéphane Brull, J. Mallet, Bruno Dubroca, Centre d'Etudes Lasers Intenses et Applications (CELIA), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), Institut de Mathématiques de Bordeaux (IMB), 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 de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Centre National de la Recherche Scientifique (CNRS), and 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)

Subjects
Physics, Angular momentum, Physics and Astronomy (miscellaneous), Discretization, Closure (topology), 010103 numerical & computational mathematics, Electron, entropic scheme, 01 natural sciences, 010101 applied mathematics, Moment (mathematics), Classical mechanics, Phase space, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Fokker–Planck equation, Statistical physics, moment systems, 0101 mathematics, Landau-Fokker-Planck equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Variable (mathematics), entropy minimization
Abstract

In plasma physics domain, the electron transport is described with the Fokker-Planck-Landau equation. The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. That is why we propose in this paper a new model whose derivation is based on an angular closure in the phase space and retains only the energy of particles as kinetic dimension. To find a solution compatible with physics conditions, the closure of the moment system is obtained under a minimum entropy principle. This model is proved to satisfy the fundamental properties like a H theorem. Moreover an entropic discretization in the velocity variable is proposed on the semi-discrete model. Finally, we validate on numerical test cases the fundamental properties of the full discrete model.

Published
2014

15. A Local Entropy Minimum Principle for Deriving Entropy Preserving Schemes

Author

Afeintou Sangam, Bruno Dubroca, Christophe Berthon, 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 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), 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), Control, Analysis and Simulations for TOkamak Research (CASTOR), 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), Laboratoire Jean Alexandre Dieudonné (LJAD), 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), 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)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), 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), and Laboratoire Jean Alexandre Dieudonné (JAD)

Subjects
[PHYS.PHYS.PHYS-FLU-DYN]Physics [physics]/Physics [physics]/Fluid Dynamics [physics.flu-dyn], Data_CODINGANDINFORMATIONTHEORY, 010103 numerical & computational mathematics, 01 natural sciences, Entropy power inequality, Entropy (information theory), Godunov type schemes, discrete entropy minimum principle, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, [MATH]Mathematics [math], Entropy rate, Mathematics, Numerical Analysis, AMS subject classifications 65M60, Applied Mathematics, Principle of maximum entropy, Mathematical analysis, Maximum entropy thermodynamics, 16. Peace & justice, discrete entropy inequalities, Quantum relative entropy, 010101 applied mathematics, Computational Mathematics, 65M12 Key words Hyperbolic system, Maximum entropy probability distribution, general Euler equations, Joint quantum entropy, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; The present work deals with the establishment of stability conditions of finite volume methods to approximate weak solutions of the general Euler equations to simulate compressible flows. In oder to ensure discrete entropy inequalities, we derive a new technique based on a local minimum principle to be satisfied by the specific entropy. Sufficient conditions are exhibited to satisfy the required local minimum entropy principle. Arguing these conditions, a class of entropy preserving schemes is thus derived.

Published
2012

16. High order resolution of the Maxwell-Fokker-Planck-Landau model intended for ICF applications

Author

Francis Filbet, Roland Duclous, Vladimir Tikhonchuk, Bruno Dubroca, Centre d'Etudes Lasers Intenses et Applications (CELIA), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), 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), ANR-06-JCJC-0136,MNEC,Méthodes Numériques pour les Equations Cinétiques(2006), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-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
Physics and Astronomy (miscellaneous), 010103 numerical & computational mathematics, 01 natural sciences, Electric field, 0103 physical sciences, FOS: Mathematics, Statistical physics, Mathematics - Numerical Analysis, 0101 mathematics, Fokker-Planck-Landau, 010306 general physics, High order numerical scheme, Physics, Numerical Analysis, Plane (geometry), Applied Mathematics, Numerical analysis, ICF, Plasma, Numerical Analysis (math.NA), Computer Science Applications, Magnetic field, Energy deposition, Computational Mathematics, Electronic transport, Distribution function, NLTE regime, Modeling and Simulation, Fokker–Planck equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Curse of dimensionality
Abstract

International audience; A high order, deterministic direct numerical method is proposed for the nonrelativistic $2D_{\bf x} \times 3D_{\bf v}$ Vlasov-Maxwell system, coupled with Fokker-Planck-Landau type operators. Such a system is devoted to the modelling of electronic transport and energy deposition in the general frame of Inertial Confinement Fusion applications. It describes the kinetics of plasma physics in the nonlocal thermodynamic equilibrium regime. Strong numerical constraints lead us to develop specific methods and approaches for validation, that might be used in other fields where couplings between equations, multiscale physics, and high dimensionality are involved. Parallelisation (MPI communication standard) and fast algorithms such as the multigrid method are employed, that make this direct approach be computationally affordable for simulations of hundreds of picoseconds, when dealing with configurations that present five dimensions in phase space.

Published
2008

17. Rarefied Pure Gas Transport in Non-isothermal Porous Media: Validation and Tests of the Model

Author

Christophe Preux, Bruno Dubroca, Gerard L. Vignoles, Pierre Charrier, Laboratoire des Composites Thermostructuraux (LCTS), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Institut de Chimie du CNRS (INC)-Snecma-SAFRAN group-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), IFP Energies nouvelles (IFPEN), Centre d'études scientifiques et techniques d'Aquitaine (CESTA), 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 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)

Subjects
Materials science, General Chemical Engineering, Thermodynamics, 02 engineering and technology, Thermal diffusivity, 01 natural sciences, Homogenization (chemistry), Catalysis, 010305 fluids & plasmas, Thermal transpiration, 0103 physical sciences, Tensor, Galerkin method, Homogenization, Kinetic equation, Mechanics, [CHIM.MATE]Chemical Sciences/Material chemistry, 021001 nanoscience & nanotechnology, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Knudsen diffusion, Heat transfer, Knudsen number, Numerical methods, 0210 nano-technology, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper presents a numerical method based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described by a digitized 3D image. The homogenization procedure -- neglecting the effect of gas density gradients on heat transfer through the solid -- leads to closure problems in R^6 for the obtention of effective properties ; they are then simplified using a Galerkin method based on a 21-element basis set. The kinetic equations are then discretized in R^3 space with a finite-volume scheme. The method is validated against experimental data in the case of a closed test tube. It shows to be coherent with past approaches of thermal transpiration. Then, it is applied to several 3D images of increasing complexity. Another validation is brought by comparison with other distinct numerical approaches for the evaluation of the Darcian permeability tensor and of the Knudsen diffusion tensor. Results show that thermal transpiration has to be described by an effective transport tensor which is distinct from the other tensors.

Published
2008

18. Rarefied Pure Gas Transport in Non-Isothermal Porous Media: Effective Transport properties from Homogenization of the Kinetic Equation

Author

Gerard L. Vignoles, Christophe Preux, Bruno Dubroca, Pierre Charrier, Laboratoire des Composites Thermostructuraux (LCTS), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Institut de Chimie du CNRS (INC)-Snecma-SAFRAN group-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), IFP Energies nouvelles (IFPEN), and 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)

Subjects
Homogenization, Materials science, General Chemical Engineering, Kinetic equation, Thermodynamics, 02 engineering and technology, [CHIM.MATE]Chemical Sciences/Material chemistry, 021001 nanoscience & nanotechnology, Thermal diffusivity, 01 natural sciences, Homogenization (chemistry), [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation, Catalysis, Isothermal process, 010305 fluids & plasmas, Thermal transpiration, Knudsen diffusion, 0103 physical sciences, Heat transfer, Knudsen number, 0210 nano-technology, Porous medium, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Viscous flow, effusion, and thermal transpiration are the main gas transport modalities for a rarefied gas in a macro-porous medium. They have been well quantified only in the case of simple geometries. This paper develops a model based on the homogenization of kinetic equations producing effective transport properties (permeability, Knudsen diffusivity, thermal transpiration ratio) in any porous medium sample, as described e. g. by a digitized 3D image. The homogenization procedure -- neglecting the effect of gas density gradients on heat transfer through the solid -- leads to macroscopic transfer relations, and to closure problems in R^6 for the obtention of effective properties. Coherence of the approach with previous literature on the subject is discussed. The asymptotic limits of the model (rarefied and continuum regimes) are also studied. One of the main results is that the effect of the geometry on thermal transpiration has to be described by a tensor which is distinct from the permeability and Knudsen diffusion tensors.

Published
2007

19. A Consistent Approach for the Coupling of Radiation and Hydrodynamics at Low Mach Number

Author

Ioan Teleaga, Mohammed Seaid, Bruno Dubroca, Institut de Mathématiques de Bordeaux (IMB), 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), Fachbereich Mathematik (FACHBEREICH MATHEMATIK), and Technische Universität Kaiserslautern (TU Kaiserslautern)

Subjects
Convection, Physics and Astronomy (miscellaneous), Discretization, 7. Clean energy, 01 natural sciences, Compressible flow, 010305 fluids & plasmas, symbols.namesake, 0103 physical sciences, Radiative transfer, 0101 mathematics, ComputingMilieux_MISCELLANEOUS, Physics, Numerical Analysis, Natural convection, Applied Mathematics, Space time, Mechanics, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Classical mechanics, Mach number, Modeling and Simulation, symbols, Compressibility, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We present a consistent numerical model for coupling radiation to hydrodynamics at low Mach number. The hydrodynamical model is based on a low-Mach asymptotic in the compressible flow that removes acoustic wave propagation while retaining the compressibility effects resulting from combustion. Radiative transfer is modelled by the M"1 entropy equations that can be viewed as a moment method. The radiation model possesses the capability to accurately approximate solution of radiative transfer at low computational cost while retaining the main physical properties of radiative energy. Consistent numerical approaches are developed for space and time discretizations in both hydrodynamics and radiation. A modified projection method is used for hydrodynamics, whereas an HLL-type discretization is implemented for radiation transport. The combined methods permit time steps that are controlled by the advective time scales resulting in a substantial improvement in computational efficiency compared to a compressible formulation. Numerical results are presented for the natural convection in a squared cavity with large temperature difference and also for a diffusion methane/air flame with four-step reduced chemical reactions in non-gray participating media. The present approach has been found to be feasible and satisfactory.

Published
2007

20. Discrete-Velocity Models for Numerical Simulations in Transitional Regime for Rarefied Flows and Radiative Transfer

Author

Luc Mieussens, Pierre Charrier, Rodolphe Turpault, and Bruno Dubroca

Subjects
Physics, ComputingMethodologies_SIMULATIONANDMODELING, 010102 general mathematics, Relaxation (iterative method), Eulerian path, 010103 numerical & computational mathematics, Gas dynamics, Kinetic energy, 01 natural sciences, Discrete velocity, symbols.namesake, Atmospheric radiative transfer codes, symbols, Radiative transfer, Statistical physics, 0101 mathematics, Minimum entropy
Abstract

In this paper we propose a deterministic eulerian approach for kinetic relaxation models. This approach is based on adefinition of the discrete equilibrium using a discrete equivalent of the minimum entropy principle. This leads to discrete models which are entropic and conservative. Two applications to gas dynamics and to radiative transfer are presented. Numerical experiments illustrate the performance of numerical codes based on this approach.

Published
2004

21. A conservative and entropic discrete-velocity model for rarefied polyatomic gases

Author

Luc Mieussens, Bruno Dubroca, Institut de Mathématiques de Bordeaux (IMB), 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 d'études scientifiques et techniques d'Aquitaine (CESTA), 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 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)

Subjects
010101 applied mathematics, Physics, 0103 physical sciences, Polyatomic ion, Statistical physics, 0101 mathematics, 01 natural sciences, Discrete velocity, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], 010305 fluids & plasmas
Abstract

International audience

Published
2001

22. A numerical approach for a system of transport equations in the field of radiotherapy

Author

Bruno Dubroca, T. Pichard, Stéphane Brull, 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 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), Laboratoire des Composites Thermostructuraux (LCTS), Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Institut de Chimie du CNRS (INC)-Snecma-SAFRAN group-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), and Centre National de la Recherche Scientifique (CNRS)-Snecma-SAFRAN group-Université de Bordeaux (UB)-Institut de Chimie du CNRS (INC)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)

Subjects
Photon, Physics and Astronomy (miscellaneous), Numerical analysis, Relaxation scheme, 010103 numerical & computational mathematics, [CHIM.MATE]Chemical Sciences/Material chemistry, Systems modeling, Kinetic energy, 01 natural sciences, Radiotherapy dose computation, 030218 nuclear medicine & medical imaging, M 1 model, 03 medical and health sciences, Discrete equation, 0302 clinical medicine, Distribution function, Realizability, Applied mathematics, M1 model, 0101 mathematics, AMS MSC : 35A35, 65M22, 35L65, 82C40, Implicit scheme, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

Numerical schemes for systems of transport equations are commonly constrained by a stability condition of Courant-Friedrichs-Lewy (CFL) type. We consider a system modeling the steady transport of photons and electrons in the field of radiotherapy. Naive discretizations of such a system are commonly constrained by a very restrictive CFL condition. This issue is circumvented by constructing an implicit scheme based on a relaxation approach.We use an entropy-based moment model, namely the M1 model. Such a system of equations possesses the non-linear flux terms of a hyperbolic system but no time derivative. The flux terms are well-defined only under a condition on the unknowns, called realizability, which corresponds to the positivity of an underlying kinetic distribution function.The present numerical approach is applicable to non-linear systems which possess no hyperbolic operator, and it preserves the realizability property. However, the discrete equations are non-linear, and we propose a numerical method to solve such non-linear systems.Our approach is tested on academic and practical cases in 1D, 2D, and 3D and it is shown to require significantly less computational power than reference methods.

23. Asymptotic-preserving well-balanced scheme for the electronic M1 model in the diffusive limit: particular cases

Author

Emmanuel d'Humières, Stéphane Brull, Bruno Dubroca, Sébastien Guisset, Centre d'Etudes Lasers Intenses et Applications (CELIA), Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Bordeaux (UB), Institut de Mathématiques de Bordeaux (IMB), 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), Laboratoire de Mathématiques de Versailles (LMV), Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Université Paris-Saclay-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), and Université de Bordeaux (UB)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Electronic M1moment model, Work (thermodynamics), Diffusion equation, diffusive limit, 010103 numerical & computational mathematics, 01 natural sciences, symbols.namesake, Electric field, Godunov type schemes, Limit (mathematics), 0101 mathematics, [MATH]Mathematics [math], Mathematics, Numerical Analysis, plasma physics, Computer Science::Information Retrieval, Applied Mathematics, Mathematical analysis, Godunov's scheme, Riemann solver, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, Scheme (mathematics), Viscosity (programming), symbols, approximate Riemann solvers, asymptotic preserving schemes, Analysis
Abstract

This work is devoted to the derivation of an asymptotic-preserving scheme for the electronic M1 model in the diffusive regime. The case without electric field and the homogeneous case are studied. The derivation of the scheme is based on an approximate Riemann solver where the intermediate states are chosen consistent with the integral form of the approximate Riemann solver. This choice can be modified to enable the derivation of a numerical scheme which also satisfies the admissible conditions and is well-suited for capturing steady states. Moreover, it enjoys asymptotic-preserving properties and handles the diffusive limit recovering the correct diffusion equation. Numerical tests cases are presented, in each case, the asymptotic-preserving scheme is compared to the classical HLL [A. Harten, P.D. Lax and B. Van Leer, SIAM Rev. 25 (1983) 35–61.] scheme usually used for the electronic M1 model. It is shown that the new scheme gives comparable results with respect to the HLL scheme in the classical regime. On the contrary, in the diffusive regime, the asymptotic-preserving scheme coincides with the expected diffusion equation, while the HLL scheme suffers from a severe lack of accuracy because of its unphysical numerical viscosity.

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