Search

Your search keyword '"ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)"' showing total 55 results
Start Over Author "ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)"
55 results on '"ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)"'

1. On the stability of totally upwind schemes for the hyperbolic initial boundary value problem

Author

Boutin, Benjamin, Barbenchon, Pierre Le, Seguin, Nicolas, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), and Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers

Subjects
Kreiss-Lopatinskii determinant, GKS stability, 65M12, 65M06, GKS-stability, finite-difference methods, boundary conditions, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], inverse Lax-Wendroff, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit totally upwind schemes in 1D with numerical boundary conditions. The underlying approximated continuous problem is the one-dimensional advection equation. The strong stability is studied using the Kreiss-Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss-Lopatinskii determinant, which possesses remarkable regularity properties. By applying standard results of complex analysis, we are able to elate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the Beam-Warming scheme together with the simplified inverse Lax-Wendroff procedure at the boundary.

Published
2023
Full Text
View/download PDF

2. Freely floating objects on a fluid governed by the Boussinesq equations

Author

Beck, Geoffrey, Lannes, David, Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), 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), Del Duca fondation, ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)

Subjects
[PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph], Free surface, Applied Mathematics, Nonlocal transport equations, Return to equilibrium, Fluid structure Interaction, Dispersive boundary layer, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], Boussinesq equations, Mathematical Physics, Analysis, Analysis of PDEs (math.AP)
Abstract

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear non dispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has an interest of its own and for which we consequently provide a general analysis., 63 pages, 2 figures

Published
2022
Full Text
View/download PDF

3. Stability of finite difference schemes for the hyperbolic initial boundary value problem by winding number computations

Author

Boutin, Benjamin, Barbenchon, Pierre Le, Seguin, Nicolas, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Montpelliérain Alexander Grothendieck (IMAG), Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), and ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011)

Subjects
Kreiss-Lopatinskii determinant, GKS-stability, FOS: Mathematics, IBVP, 65M12, 65M06, 30E10, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, winding number, finite difference methods, MSC: 65M12, 65M06, 30E10, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper, we present a numerical strategy to check the strong stability (or GKS-stability) of one-step explicit finite difference schemes for the one-dimensional advection equation with an inflow boundary condition. The strong stability is studied using the Kreiss-Lopatinskii theory. We introduce a new tool, the intrinsic Kreiss-Lopatinskii determinant, which possesses the same regularity as the vector bundle of discrete stable solutions. By applying standard results of complex analysis to this determinant, we are able to relate the strong stability of numerical schemes to the computation of a winding number, which is robust and cheap. The study is illustrated with the O3 scheme and the fifth-order Lax-Wendroff (LW5) scheme together with a reconstruction procedure at the boundary., arXiv admin note: text overlap with arXiv:2207.10978

Published
2023

4. Sharp stability for finite difference approximations of hyperbolic equations with boundary conditions

Author

Coulombel, Jean-François, Faye, Grégory, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Applied Mathematics, General Mathematics, Numerical Analysis (math.NA), stability, hyperbolic equations, semigroup estimates, Computational Mathematics, Mathematics - Analysis of PDEs, AMS classification: 65M06, 65M12, 47B35, 35L04, 35L20, boundary conditions, difference approximations, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Toeplitz operators, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

In this article, we consider a class of finite rank perturbations of Toeplitz operators that have simple eigenvalues on the unit circle. Under a suitable assumption on the behavior of the essential spectrum, we show that such operators are power bounded. The problem originates in the approximation of hyperbolic partial differential equations with boundary conditions by means of finite difference schemes. Our result gives a positive answer to a conjecture by Trefethen, Kreiss and Wu that only a weak form of the so-called uniform Kreiss–Lopatinskii condition is sufficient to imply power boundedness.

Published
2021
Full Text
View/download PDF

5. Développement d'un modèle dispersif hyperbolique pour les vagues côtières et implémentation dans Tolosa. Derivation of a disspersive-hyperbolic model for coastal waves et its numerical implementation

Author

Duran, Arnaud, Kazakova, Maria, Hung, Yen-Chung, Baraille, Rémy, Couderc, Frédéric, Vila, Jean-Paul, Chauchat, Julien, Richard, Gaël Loïc, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques (LAMA), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS), SHOM SERVICE HYDROGRAPHIQUE ET OCEANOGRAPHIQUE DE LA MARINE BREST FRA, Partenaires IRSTEA, Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées, Laboratoire des Écoulements Géophysiques et Industriels [Grenoble] (LEGI), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Erosion torrentielle neige et avalanches (UR ETGR (ETNA)), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In the present study we propose a new hyperbolic model with the same dispersive properties as the classical Serre-Green-Naghdi capable to capture wave breaking phenomenon.; Dans cette étude, nous proposons un nouveau modèle hyperbolique avec les mêmes propriétés dispersives que le modèle classique de Serre-Green-Naghdi capable de capturer le phénomène de déferlement des vagues.

Published
2022

6. Développement d'un modèle dispersif hyperbolique pour les vagues côtières et implémentation dans Tolosa

Author

Duran, Arnaud, Kazakova, Maria, Hung, Yen-Chung, Baraille, Rémy, Couderc, Frédéric, Vila, Jean-Paul, Chauchat, Julien, Richard, Gaël Loïc, 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), 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), Laboratoire de Mathématiques (LAMA), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS), SHOM SERVICE HYDROGRAPHIQUE ET OCEANOGRAPHIQUE DE LA MARINE BREST FRA, Partenaires IRSTEA, Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT), Laboratoire des Écoulements Géophysiques et Industriels [Grenoble] (LEGI), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Erosion torrentielle neige et avalanches (UR ETGR (ETNA)), Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
[SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph]
Abstract

International audience; In this study, we propose a new hyperbolic model with the same dispersive properties as the classical Serre-Green-Naghdi model capable to capture the wave breaking phenomenon. The hyperbolisation of the equations is based on taking into account the finite character of the sound speed and thus the compressibility of the sea water. The method therefore involves introducing acoustic energy into the system. In the case of coastal waves, the effects of static compressibility are negligible. The resulting model is then a hyperbolic approximation of the Serre-Green-Naghdi equations. The modelling of breakingwaves is obtained by adapting to this pseudo-compressible approach the depth-averaging method of Large-Eddy Simulations (LES) where the small scale turbulence is modelled by a turbulent viscosity, whereas the large scales are taken into account in the model by an anisotropic tensor variable called enstrophy.; Dans cette étude, nous proposons un nouveau modèle hyperbolique avec les mêmes propriétés dispersives que le modèle classique de Serre-Green-Naghdi capable de capturer le phénomène de déferlement des vagues. L'hyperbolisation des équations est basée sur la prise en compte du caractère fini de la vitesse du son et donc de la compressibilité de l'eau de mer. La méthode implique donc d'introduire une énergie acoustique dans le système. Dans le cas des vagues côtières, les effets de compressibilité statique sont négligeables. Le modèle obtenu est alors une approximation hyperbolique des équations de Serre-Green-Naghdi. La modélisation du déferlement est obtenue par l'adaptation à cette approche pseudo-compressible de la méthode de moyenne sur la profondeur des équations de la simulation des grandes échelles de la turbulence (SGS) où les petites échelles de la turbulence sont modélisées par une viscosité turbulente, en revanche les grandes échelles sont prises en compte dans le modèle par une variable tensorielle anisotrope appelée enstrophie.

Published
2022

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

Author

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

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

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

Published
2020
Full Text
View/download PDF

8. Local limit theorem for complex valued sequences

Author

Coeuret, Lucas, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-21-CE40-0008,Indyana,Dynamiques d'invasion et asymptotiques non triviales(2021), 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 Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
local limit theorem, difference approximation, Probability (math.PR), 42A85, 35K25, 60F99, 65M12, Numerical Analysis (math.NA), stability, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], FOS: Mathematics, AMS classification MSC: 42A85, 35K25, 60F99, 65M12, convolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics - Probability
Abstract

In this article, we study the pointwise asymptotic behavior of iterated convolutions on the one dimensional lattice Z. We generalize the so-called local limit theorem in probability theory to complex valued sequences. A sharp rate of convergence towards an explicitly computable attractor is proved together with a generalized Gaussian bound for the asymptotic expansion up to any order of the iterated convolution.

Published
2022

9. The Green's function of the Lax-Wendroff and Beam-Warming schemes

Author

Coulombel, Jean-François, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
difference approximation, local limit theorem, Lax-Wendroff scheme, [MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA], Numerical Analysis (math.NA), stability, MSC : 65M06, 65M12, 35L02, Beam-Warming scheme, Mathematics - Classical Analysis and ODEs, transport equation, Classical Analysis and ODEs (math.CA), FOS: Mathematics, convolution, Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We prove a sharp uniform generalized Gaussian bound for the Green's function of the Lax-Wendroff and Beam-Warming schemes. Our bound highlights the spatial region that leads to the well-known (rather weak) instability of these schemes in the maximum norm. We also recover uniform bounds in the maximum norm when these schemes are applied to initial data of bounded variation.

Published
2022
Full Text
View/download PDF

10. The Neumann boundary condition for the two-dimensional Lax-Wendroff scheme

Author

Coulombel, Jean-François, Benoit, Antoine, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), Université du Littoral Côte d'Opale (ULCO), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Research of the author was supported by ANR project NABUCO, ANR-17-CE40-0025, and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
numerical schemes, boundary conditions, transport equations, FOS: Mathematics, Mathematics - Numerical Analysis, domains with corners, Numerical Analysis (math.NA), stability, MSC: 65M12, 65M06, 65M20, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We study the stability of the two-dimensional Lax-Wendroff scheme with a stabilizer that approximates solutions to the transport equation. The problem is first analyzed in the whole space in order to show that the so-called energy method yields an optimal stability criterion for this finite difference scheme. We then deal with the case of a half-space when the transport operator is outgoing. At the numerical level, we enforce the Neumann extrapolation boundary condition and show that the corresponding scheme is stable. Eventually we analyze the case of a quarter-space when the transport operator is outgoing with respect to both sides. We then enforce the Neumann extrapolation boundary condition on each side of the boundary and propose an extrapolation boundary condition at the numerical corner in order to maintain stability for the whole numerical scheme.

Published
2022
Full Text
View/download PDF

11. Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with Dirichlet and oblique boundary conditions

Author

Guillaume Bertoli, Christophe Besse, Gilles Vilmart, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diffusion equation, Nonhomo- geneous boundary conditions, order reduction, Mathematics::Numerical Analysis, Mathematics - Analysis of PDEs, Convergence (routing), FOS: Mathematics, Applied mathematics, Crank–Nicolson method, AMS subject classifications (2020): 65M12, 65L04, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Crank-Nicolson, Mathematics - Numerical Analysis, Boundary value problem, ddc:510, Order reduction, Mathematics, diffusion-reaction equation, Algebra and Number Theory, Diffusion-reaction equation, Applied Mathematics, Oblique case, Numerical Analysis (math.NA), Superconvergence, nonhomogeneous boundary conditions, Computational Mathematics, Strang splitting, Flow (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We prove these results when the source term only depends on the space variable, an assumption which makes the splitting scheme equivalent to the Crank-Nicolson method itself applied to the whole problem. Numerical experiments suggest that the second order convergence persists with general nonlinearities.

Published
2021

12. Lower exponential strong well-posedness of hyperbolic boundary value problems in a strip

Author

Antoine Benoit, Université du Littoral Côte d'Opale (ULCO), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
General Mathematics, 010102 general mathematics, 0103 physical sciences, Mathematical analysis, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, Boundary value problem, 0101 mathematics, 01 natural sciences, Well posedness, Mathematics, Exponential function
Abstract

International audience; In this article we are interested in the lower exponential strong well-posedness of hyperbolic boundary value problems set in the strip R d−1 × [0, 1]. We assume that each component of the boundary of the strip satisfies the so-called uniform Kreiss-Lopatinskii condition (which is the condition ensuring the lower exponential strong well-posedness of the boundary value problem in the half space) and we show that due to selfinteraction phenomenon, an extra condition has to be made to ensure the lower exponential strong well-posedness of the boundary value problem in the strip. This condition imposes that a selfinteracting wave in the strip can not be amplified by repetitive reflections against each component of the boundary. This condition is very analogous to the one imposed in [Osher, 1973] to study the problem in the quarter space. However, due to the simplicity of the phases generation process for the strip geometry (less phases are generated) and this condition involves matrices and not Fourier integral operators.

Published
2020
Full Text
View/download PDF

13. The Neumann numerical boundary condition for transport equations

Author

Frédéric Lagoutière, Jean-François Coulombel, Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Institut de Mathématiques de Toulouse UMR5219 ( IMT ), Université Toulouse 1 Capitole ( UT1 ) -Université Toulouse - Jean Jaurès ( UT2J ) -Université Toulouse III - Paul Sabatier ( UPS ), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-PRES Université de Toulouse-Institut National des Sciences Appliquées - Toulouse ( INSA Toulouse ), Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Centre National de la Recherche Scientifique ( CNRS ), Institut Camille Jordan [Villeurbanne] ( ICJ ), Centre National de la Recherche Scientifique ( CNRS ) -Université Jean Monnet [Saint-Étienne] ( UJM ) -École Centrale de Lyon ( ECL ), Université de Lyon-Université de Lyon-Institut National des Sciences Appliquées de Lyon ( INSA Lyon ), Université de Lyon-Institut National des Sciences Appliquées ( INSA ) -Institut National des Sciences Appliquées ( INSA ) -Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon, ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion. ( 2013 ), ANR-17-CE40-0025,NABUCO,NumericAl BoUndaries and COupling, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), and Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Discretization, Type (model theory), 01 natural sciences, Stability (probability), 010305 fluids & plasmas, Mathematics - Analysis of PDEs, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Normal mode, numerical schemes, 0103 physical sciences, Convergence (routing), Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Mathematics, Numerical Analysis, convergence, Mathematical analysis, transport equations, [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], stability, 010101 applied mathematics, Modeling and Simulation, Outflow boundary, AMS classification: 65M12, 65M06, 65M20, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this article, we show that prescribing homogeneous Neumann type numerical boundary conditions at an outflow boundary yields a convergent discretization in \begin{document}$ \ell^\infty $\end{document} for transport equations. We show in particular that the Neumann numerical boundary condition is a stable, local, and absorbing numerical boundary condition for discretized transport equations. Our main result is proved for explicit two time level numerical approximations of transport operators with arbitrarily wide stencils. The proof is based on the energy method and bypasses any normal mode analysis.

Published
2020
Full Text
View/download PDF

14. Numerical approximation of the shallow water equations with Coriolis source term

Author

Yohan Penel, Arnaud Duran, Youssouf Nasseri, Noémie Gaveau, Virgile Dubos, Emmanuel Audusse, Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, 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 (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), 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), 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), Institut Denis Poisson (IDP), Université d'Orléans (UO)-Université de Tours (UT)-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), The authors express their deep gratitude to the French National Research Agency project NABUCO, grant ANR-17-CE40-0025 and the french INSU-CNRS (Institut National des Sciences de l’Univers – Centre National de la RechercheScientifique) program LEFE-MANU (Méthodes Math ématiques et Numériques), project DWAVE for the funding of the project, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), Institut Camille Jordan [Villeurbanne] (ICJ), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), and Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU)

Subjects
T57-57.97, Work (thermodynamics), Applied mathematics. Quantitative methods, Finite volume method, Mathematical analysis, Term (time), symbols.namesake, QA1-939, Froude number, symbols, Polygon mesh, Diffusion (business), [MATH]Mathematics [math], Shallow water equations, Mathematics, Geostrophic wind, Physics::Atmospheric and Oceanic Physics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We investigate in this work a class of numerical schemes dedicated to the non-linear Shallow Water equations with topography and Coriolis force. The proposed algorithms rely on Finite Volume approximations formulated on collocated and staggered meshes, involving appropriate diffusion terms in the numerical fluxes, expressed as discrete versions of the linear geostrophic balance. It follows that, contrary to standard Finite-Volume approaches, the linear versions of the proposed schemes provide a relevant approximation of the geostrophic equilibrium. We also show that the resulting methods ensure semi-discrete energy estimates. Numerical experiments exhibit the efficiency of the approach in the presence of Coriolis force close to the geostrophic balance, especially at low Froude number regimes.

Published
2021
Full Text
View/download PDF

15. ABOUT PLANE PERIODIC WAVES OF THE NONLINEAR SCHRÖDINGER EQUATIONS

Author

Audiard, Corentin, Rodrigues, L. Miguel, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), 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 Universitaire de France (IUF), Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche (M.E.N.E.S.R.), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), 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 Rennes Angers, 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
Hamiltonian dynamics 35B10, periodic traveling waves, orbital stability, soliton asymptotics, 37K45, Schrödinger equations, 35B35, 35Q55, spectral stability, modulation systems, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], abbreviated action integral, 35P05, harmonic limit
Abstract

International audience; The present contribution contains a quite extensive theory for the stability analysis of plane periodic waves of general Schrödinger equations. On one hand, we put the one-dimensional theory, or in other words the stability theory for longitudinal perturbations, on a par with the one available for systems of Korteweg type, including results on co-periodic spectral instability, nonlinear co-periodic orbital stability, side-band spectral instability and linearized large-time dynamics in relation with modulation theory, and resolutions of all the involved assumptions in both the small-amplitude and large-period regimes. On the other hand, we provide extensions of the spectral part of the latter to the multi-dimensional context. Notably, we provide suitable multi-dimensional modulation formal asymptotics, validate those at the spectral level and use them to prove that waves are always spectrally unstable in both the small-amplitude and the large-period regimes.; Cet article apporte des résultats assez larges sur la stabilité des ondes planes périodiques d'une classe d'équation de type Schrödinger. D'une part, on amène la théorie en dimension un (ou autrement dit, pour les perturbations longitudinales) au niveau de celle existant pour les systèmes de type Korteweg ; notamment les résultats d'instabilité spectrale copériodique, stabilité orbitale copériodique, instabilité modulationnelle, dynamique linéarisée en temps long modulationnelle, ainsi que la justification rigoureuse de toutes les hypothèses dans les régimes de petite amplitude et de grande période. D'autre part, on généralise la partie spectrale des résultats précédents en dimension supérieure à un. En particulier on obtient formellement un système de modulation qu'on relie rigoureusement à la stabilité spectrale, et on prouve l'instabilité transverse instables des ondes dans les régimes petite amplitude et longue période.

Published
2021
Full Text
View/download PDF

16. Numerical Simulations on Nonlinear Quantum Graphs with the GraFiDi Library

Author

Besse, Christophe, Duboscq, Romain, Le Coz, Stefan, 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), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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
Statistics and Probability, Numerical Analysis, FOS: Physical sciences, Numerical Analysis (math.NA), Pattern Formation and Solitons (nlin.PS), Nonlinear Sciences - Pattern Formation and Solitons, Ground states, Computational Mathematics, Mathematics - Analysis of PDEs, [NLIN.NLIN-PS]Nonlinear Sciences [physics]/Pattern Formation and Solitons [nlin.PS], Quantum Graphs, Modeling and Simulation, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Finite Differences, Mathematics - Numerical Analysis, Python Library, Nonlinear Schrodinger equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; Nonlinear quantum graphs are metric graphs equipped with a nonlinear Schrödinger equation. Whereas in the last ten years they have known considerable developments on the theoretical side, their study from the numerical point of view remains in its early stages. The goal of this paper is to present the Grafidi library, a Python library which has been developed with the numerical simulation of nonlinear Schrödinger equations on graphs in mind. We will show how, with the help of the Grafidi library, one can implement the popular normalized gradient flow and nonlinear conjugate gradient flow methods to compute ground states of a nonlinear quantum graph. We will also simulate the dynamics of the nonlinear Schrödinger equation with a Crank-Nicolson relaxation scheme and a Strang splitting scheme. Finally, in a series of numerical experiments on various types of graphs, we will compare the outcome of our numerical calculations for ground states with the existing theoretical results, thereby illustrating the versatility and efficiency of our implementations in the framework of the Grafidi library.

Published
2021
Full Text
View/download PDF

17. Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators

Author

Joackim Bernier, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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), 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), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Quadratic differential operators, Applied Mathematics, Numerical analysis, Splitting methods, Numerical Analysis (math.NA), 010103 numerical & computational mathematics, 01 natural sciences, Schrödinger equation, Computational Mathematics, symbols.namesake, Mathematics - Analysis of PDEs, Quadratic equation, Computational Theory and Mathematics, Factorization, FOS: Mathematics, symbols, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Quadratic differential, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that can be approximated efficiently, using, for example, pseudo-spectral methods. We highlight the efficiency of these new methods on the examples of the magnetic linear Schrödinger equations with quadratic potentials, some transport equations and some Fokker-Planck equations.

Published
2021

18. Derivation and analysis of computational methods for fractional Laplacian equations with absorbing layers

Author

Yong Zhang, Emmanuel Lorin, Xavier Antoine, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), School of Mathematics and Statistics [Carleton University], Carleton University, Tianjin University (TJU), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
fractional partial differential equations, Fourier pseudospectral approximation, Applied Mathematics, Numerical analysis, Computation, Finite difference, 010103 numerical & computational mathematics, Space (mathematics), finite-difference, 01 natural sciences, time splitting scheme, 010101 applied mathematics, symbols.namesake, Fourier transform, Theory of computation, symbols, Applied mathematics, perfectly matched layers, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Pseudo-spectral method, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; This paper is devoted to the derivation and analysis of accurate and efficient Perfectly Matched Layers (PML) or efficient absorbing layers for solving fractional Laplacian equations within Initial Boundary Value Problems (IBVP). Two main approaches are derived: we first propose a Fourier-based pseu-dospectral method, and then present a real space method based on an efficient computation of the fractional Laplacian with PML. Some numerical experiments and analytical results are proposed along the paper to illustrate the presented methods.

Published
2021
Full Text
View/download PDF

19. Exact Splitting Methods for Kinetic and Schrödinger Equations

Author

Joackim Bernier, Nicolas Crouseilles, Yingzhe Li, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Multi-scale numerical geometric schemes (MINGUS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), 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)-Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université de Rennes (UR), Inria Rennes – Bretagne Atlantique, Institut National de Recherche en Informatique et en Automatique (Inria), University of Chinese Academy of Sciences [Beijing] (UCAS), Laboratory of Scientific and Engineering Computing [Beijing] (LSEC), Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences [Beijing] (CAS)-Chinese Academy of Sciences [Beijing] (CAS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), 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), 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)-AGROCAMPUS OUEST, Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Inria Rennes – Bretagne Atlantique, and Université de Rennes (UNIV-RENNES)

Subjects
Angular momentum, Quadratic differential operators, Space (mathematics), 01 natural sciences, Theoretical Computer Science, Schrödinger equation, symbols.namesake, Mathematics - Numerical Analysis, 0101 mathematics, Quadratic differential, Mathematics, Mathematical physics, Numerical Analysis, Applied Mathematics, Numerical analysis, General Engineering, Splitting methods, Rotating Bose Einstein condensate, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Computational Theory and Mathematics, symbols, Fokker–Planck equation, Fokker-Planck, Rotation (mathematics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Software
Abstract

In (Bernier in Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators. arXiv:1912.13219 , (2019)), some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, Fokker–Planck equations, and Schrodinger type equations with an angular momentum rotation term. In this work, these exact splittings are used combined with pseudo-spectral methods in space. High accuracy and efficiency of exact splitting methods are illustrated and comparison are performed with the numerical methods in literature. We show that our methods can be used to improve significantly some classical splitting methods for some nonlinear or non-quadratic equations.

Published
2021
Full Text
View/download PDF

20. High order numerical schemes for transport equations on bounded domains

Author

Jean-François Coulombel, Thi Hoai Thuong Nguyen, Abraham Sylla, Benjamin Boutin, Sébastien Tran-Tien, 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 Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO), École normale supérieure - Lyon (ENS Lyon), 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), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), École normale supérieure de Lyon (ENS de Lyon), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO)

Subjects
T57-57.97, Applied mathematics. Quantitative methods, Finite difference, Extrapolation, Boundary (topology), Numerical Analysis (math.NA), Boundary layer, Rate of convergence, Bounded function, FOS: Mathematics, QA1-939, Applied mathematics, Boundary value problem, Mathematics - Numerical Analysis, Convection–diffusion equation, Mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.

Published
2021

21. Artificial boundary conditions for the semi-discretized one-dimensional nonlocal Schrödinger equation

Author

Gang Pang, Songsong Ji, Xavier Antoine, Jiwei Zhang, College of Engineering [Beijing], Peking University [Beijing], Beijing University of Aeronautics and Astronautics (BUAA), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), School of Mathematics and Statistics [Wuhan University], Wuhan University [China], This research is partially supported by NSFC under grant Nos. NSFC 11832001, 11502028,11771035, and the Natural Science Foundation of Hubei Province No. 2019CFA007 and Xiangtan University 2018ICIP01, and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
General method, Physics and Astronomy (miscellaneous), Discretization, 010103 numerical & computational mathematics, 01 natural sciences, Schrödinger equation, symbols.namesake, semi-discrete scheme, nonlocal Schrödinger equation, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Mathematics, Numerical Analysis, Applied Mathematics, Numerical analysis, Function (mathematics), Computer Science Applications, 010101 applied mathematics, Computational Mathematics, transparent boundary condition, Modeling and Simulation, symbols, artificial boundary condition, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; A general method is proposed to build exact artificial boundary conditions for the one-dimensional nonlocal Schrödinger equation. To this end, we first consider the spatial semi-discretization of the nonlocal equation, and then develop an accurate numerical method for computing the Green's function of the semi-discrete nonlocal Schrödinger equation. These Green's functions are next used to build the exact boundary conditions corresponding to the semi-discrete model. Numerical results illustrate the accuracy of the boundary conditions. The methodology can also be applied to other nonlocal models and could be extended to higher dimensions.

Published
2021
Full Text
View/download PDF

22. Generalized gaussian bounds for discrete convolution powers

Author

Coulombel, Jean-François, Faye, Grégory, 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), ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and ANR-21-CE40-0008,Indyana,Dynamiques d'invasion et asymptotiques non triviales(2021)

Subjects
difference approximation, local limit theorem, General Mathematics, Probability (math.PR), Numerical Analysis (math.NA), stability, AMS classification MSC: 42A85, 35K25, 60F99, 65M12, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], FOS: Mathematics, convolution, Mathematics - Numerical Analysis, AMS classification MSC: 42A85, 65M12, 35K25, 60F99, Mathematics - Probability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; We prove a uniform generalized gaussian bound for the powers of a discrete convolution operator in one space dimension. Our bound is derived under the assumption that the Fourier transform of the coefficients of the convolution operator is a trigonometric rational function, which generalizes previous results that were restricted to trigonometric polynomials. We also allow the modulus of the Fourier transform to attain its maximum at finitely many points over a period.

Published
2021

23. WAVES INTERACTING WITH A PARTIALLY IMMERSED OBSTACLE IN THE BOUSSINESQ REGIME

Author

Didier Bresch, Guy Métivier, David Lannes, Laboratoire de Mathématiques (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), 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), Conseil Régional Aquitaine, Fondation Del Duca, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), and ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018)

Subjects
Free surface, Motion (geometry), Perturbation (astronomy), FOS: Physical sciences, Scale (descriptive set theory), Wave-structure interaction, Physics - Classical Physics, 01 natural sciences, Local well posedness, Boussinesq system, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Oscillations in time, Mathematics, Numerical Analysis, Transmission problem, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Classical Physics (physics.class-ph), Compatibility conditions, Dispersive boundary layer, 010101 applied mathematics, Boundary layer, Transmission (telecommunications), Focus (optics), Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for 2 × 2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a configuration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors' knowledge, this is the first time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.

Published
2021

24. Discrete transparent boundary conditions for the two-dimensional leap-frog scheme: approximation and fast implementation

Author

Christophe Besse, Jean-François Coulombel, Pascal Noble, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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), and Institut National des Sciences Appliquées (INSA)

Subjects
Coupling, Numerical Analysis, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Finite difference, leap-frog schemes, stability, 01 natural sciences, Domain (mathematical analysis), Regular grid, 010101 applied mathematics, Computational Mathematics, Transport equation, Normal mode, Modeling and Simulation, Rectangle, Boundary value problem, 0101 mathematics, [MATH]Mathematics [math], Convection–diffusion equation, transparent boundary conditions, Analysis, Mathematics
Abstract

International audience; We develop a general strategy in order to implement approximate discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leap-frog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle is analyzed by means of the so-called normal mode analysis. Numerical investigations for the full problem on the rectangle show that strong instabilities may occur when coupling stable strategies on each side of the rectangle. Other coupling strategies yield promising results.

Published
2021
Full Text
View/download PDF

25. Discrete transparent boundary conditions for the two-dimensional leap-frog scheme

Author

Besse, Christophe, Coulombel, Jean-François, NOBLE, Pascal, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, Numerical Analysis (math.NA), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; We develop a general strategy in order to implement (approximate) discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leapfrog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle is analyzed by means of the so-called normal mode analysis. Numerical investigations for the full problem on the rectangle show that strong instabilities may occur when coupling stable strategies on each side of the rectangle. Other coupling strategies yield promising results.

Published
2021

26. Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with oblique boundary conditions

Author

Bertoli, Guillaume, Besse, Christophe, Vilmart, Gilles, 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), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
diffusion-reaction equation, AMS subject classifications (2020): 65M12, 65L04, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Crank-Nicolson, order reduction, nonhomogeneous boundary conditions, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics::Numerical Analysis, Strang splitting
Abstract

We show that the Strang splitting method applied to a diffusion-reaction equation with inhomogeneous general oblique boundary conditions is of order two when the diffusion equation is solved with the Crank-Nicolson method, while order reduction occurs in general if using other Runge-Kutta schemes or even the exact flow itself for the diffusion part. We also show that this method recovers stationary states in contrast with splitting methods in general. We prove these results when the source term only depends on the space variable. Numerical experiments suggest that the second order convergence persists with general nonlinearities.

Published
2020

27. The Leray-G{\aa}rding method for finite difference schemes. II. Smooth crossing modes

Author

Coulombel, Jean-François, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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), and Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
MSC : 65M06, 65M12, 35L03, 35L04, boundary conditions, difference approximations, multiplier, Mathematics - Numerical Analysis, stability, hyperbolic equations, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], semigroup estimates
Abstract

In [Cou15] a multiplier technique, going back to Leray and G{\aa}rding for scalar hyperbolic partial differential equations, has been extended to the context of finite difference schemes for evolutionary problems. The key point of the analysis in [Cou15] was to obtain a discrete energy-dissipation balance law when the initial difference operator is multiplied by a suitable quantity. The construction of the energy and dissipation functionals was achieved in [Cou15] under the assumption that all modes were separated. We relax this assumption here and construct, for the same multiplier as in [Cou15], the energy and dissipation functionals when some modes cross. Semigroup estimates for fully discrete hy-perbolic initial boundary value problems are deduced in this broader context by following the arguments of [Cou15]., Comment: arXiv admin note: text overlap with arXiv:1505.06060

Published
2020

28. Gradient Flow Approach to the Calculation of Ground States on Nonlinear Quantum Graphs

Author

Besse, Christophe, Duboscq, Romain, Coz, Stefan Le, 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), Institut National des Sciences Appliquées (INSA), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-11-IDEX-0002,UNITI,Institut for Advanced Study in Toulouse(2011), ANR-14-CE25-0009,MAToS,Analyse des singularités topologiques dans quelques problèmes issus de la physique mathématique(2014), 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), ANR-11-IDEX-0002,UNITI,Université Fédérale de Toulouse(2011), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)

Subjects
Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Ocean Engineering, Numerical Analysis (math.NA), Mathematics - Numerical Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP)
Abstract

International audience; We introduce and implement a method to compute stationary states of nonlinear Schr\"odinger equations on metric graphs. Stationary states are obtained as local minimizers of the nonlinear Schr\"odinger energy at fixed mass. Our method is based on a normalized gradient flow for the energy (i.e. a gradient flow projected on a fixed mass sphere) adapted to the context of nonlinear quantum graphs. We first prove that, at the continuous level, the normalized gradient flow is well-posed, mass-preserving, energy diminishing and converges (at least locally) towards stationary states. We then establish the link between the continuous flow and its discretized version. We conclude by conducting a series of numerical experiments in model situations showing the good performance of the discrete flow to compute stationary states. Further experiments as well as detailed explanation of our numerical algorithm are given in a companion paper.

Published
2020
Full Text
View/download PDF

29. Long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations

Author

Benoît Grébert, Joackim Bernier, 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), 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), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-15-CE40-0001,BEKAM,Au-delà de la théorie KAM(2015), ANR-16-CE40-0013,ISDEEC,Interactions entre Systèmes Dynamiques, Equations d'Evolution et Contrôle(2016), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), and Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS)

Subjects
rational normal forms, Pure mathematics, Integrable system, resonances, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Open set, Mathematics::Analysis of PDEs, Dynamical Systems (math.DS), 01 natural sciences, Mathematics (miscellaneous), Mathematics - Analysis of PDEs, 0103 physical sciences, dispersive equations, Birkhoff normal forms, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Remainder, Mathematics - Dynamical Systems, Mathematics, Mechanical Engineering, 010102 general mathematics, Order (ring theory), 16. Peace & justice, Sobolev space, Norm (mathematics), Vector field, 010307 mathematical physics, Constant (mathematics), Analysis, Analysis of PDEs (math.AP)
Abstract

We provide an accurate description of the long time dynamics of the solutions of the generalized Korteweg–De Vries (gKdV) and Benjamin–Ono (gBO) equations on the one dimension torus, without external parameters, and that are issued from almost any (in probability and in density) small and smooth initial data. In particular, we prove a long-time stability result in Sobolev norm: given a large constant r and a sufficiently small parameter $$\varepsilon $$ , for generic initial datum u(0) of size $$\varepsilon $$ , we control the Sobolev norm of the solution u(t) for times of order $$\varepsilon ^{-r}$$ . These results are obtained by putting the system in rational normal form: we conjugate, up to some high order remainder terms, the vector fields of these equations to integrable ones on large open sets surrounding the origin in high Sobolev regularity. We stress out that our normal form technics allow to deal, for the first time, with unbounded nonlinearities containing terms of even order.

Published
2020

30. Pseudodifferential adiabatic mode parabolic equations in curvilinear coordinates and their numerical solution

Author

Pavel S. Petrov, Xavier Antoine, Il'ichev Pacific oceanological institute (POI), Il'ichev Pacific Oceanological Institute, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), The reported study was accomplished during P.S. Petrov’s visit to the Institut Elie Cartan de Lorraine (Université de Lorraine, France) supportedby the French government in the framework of the Metchnikov 2019 program. P.S. Petrov was also partly supported by the Russian Foundation for Basic Research (RFBR) under the contracts No. 18-35-20081 mol a ved and No. 18-05-00057 a. X. Antoine was supported by the French NationalResearch Agency project NABUCO, grant ANR-17-CE40-0025., and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Physics and Astronomy (miscellaneous), Physics::Optics, computational ocean acoustics, 010103 numerical & computational mathematics, 01 natural sciences, Wedge (geometry), multiple-scale method, Padé approximant, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], PACS: 43.30.+m , 43.20.Bi, 0101 mathematics, parabolic equation method, Adiabatic process, Physics, Numerical Analysis, Curvilinear coordinates, curvilinear coordinates, Whispering gallery, Applied Mathematics, Mathematical analysis, Parabolic partial differential equation, Computer Science Applications, 010101 applied mathematics, Computational Mathematics, Inflection point, Modeling and Simulation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; Pseudodifferential mode parabolic equations in curvilinear coordinates are derived from the horizontal refraction equations. An energy-conserving split-step Padé method for their numerical solution is developed. Several examples are considered including the propagation of sound in a perfect wedge, a whispering gallery formation near circular isobath, and the breaking of the whispering gallery waveguide in the vicinity of an inflection point of the isobath.

Published
2020
Full Text
View/download PDF

31. MODELING SHALLOW WATER WAVES

Author

David Lannes, 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), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), and ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018)

Subjects
General Physics and Astronomy, FOS: Physical sciences, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 14. Life underwater, [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Korteweg–de Vries equation, Mathematical Physics, Physics::Atmospheric and Oceanic Physics, Mathematics, [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Scalar (physics), Fluid Dynamics (physics.flu-dyn), Breaking wave, Statistical and Nonlinear Physics, Physics - Fluid Dynamics, Vorticity, Euler equations, 010101 applied mathematics, Nonlinear system, Waves and shallow water, Physics - Atmospheric and Oceanic Physics, 13. Climate action, Free surface, Atmospheric and Oceanic Physics (physics.ao-ph), symbols, Analysis of PDEs (math.AP)
Abstract

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the "turbulent" and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre-Green-Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe-Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension $d=1$; among them are the KdV, BBM, Camassa-Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents., Comment: 57 pages, to appear in Nonlinearity

Published
2020

32. Perfectly Matched Layer for computing the dynamics of nonlinear Schrödinger equations by pseudospectral methods. Application to rotating Bose-Einstein condensates

Author

Christophe Geuzaine, Qinglin Tang, Xavier Antoine, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Department of Electrical Engineering and Computer Science (Institut Montefiore), Université de Liège, Sichuan University [Chengdu] (SCU), The authors acknowledge the support from the Inria associate team BEC2HPC (Bose-Einstein Conden-sates : Computation and HPC simulation)., X. Antoine thanks the French National Research Agency project NABUCO grant No.ANR-17-CE40-0025., Q. Tang thanks the support of the Fundamental Research Fundsfor the Central Universities (YJ201807), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Discretization, 010103 numerical & computational mathematics, 01 natural sciences, Schrödinger equation, symbols.namesake, Lagrangian and Eulerian specification of the flow field, rotating Gross-Pitaevskii equation, 0103 physical sciences, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 010306 general physics, nonlinear Schrödinger equation, time-splitting scheme, Nonlinear Schrödinger equation, Physics, Numerical Analysis, Bose-Einstein condensate, Applied Mathematics, Relaxation (iterative method), perfectly matched layers (PML), Nonlinear system, Fourier pseudospectral method, Transformation (function), Perfectly matched layer, Modeling and Simulation, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

In this paper, we first propose a general strategy to implement the Perfectly Matched Layer (PML) approach in the most standard numerical schemes used for simulating the dynamics of nonlinear Schrodinger equations. The methods are based on the time-splitting Bao et al. (2003)[15] or relaxation Besse (2004)[24] schemes in time, and FFT-based pseudospectral discretization method in space. A thorough numerical study is developed for linear and nonlinear problems to understand how the PML approach behaves (absorbing function and tuning parameters) for a given scheme. The extension to the rotating Gross-Pitaevskii equation is then proposed by using the rotating Lagrangian coordinates transformation method Antonelli et al. (2012), Bao et al. (2013), Garcia-Ripoll et al. (2001)[13,16,38], some numerical simulations illustrating the strength of the proposed approach.

Published
2020
Full Text
View/download PDF

33. On some stable boundary closures of finite difference schemes for the transport equation

Author

Coulombel, Jean-François, Lundquist, Tomas, 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), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), 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
numerical schemes, extrapolation boundary condition, transport equation, FOS: Mathematics, Mathematics - Numerical Analysis, Numerical Analysis (math.NA), stability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], energy
Abstract

We explore in this article the possibilities and limitations of the so-called energy method for analyzing the stability of finite difference approximations to the transport equation with extrapolation numerical boundary conditions at the outflow boundary. We first show that for the most simple schemes, namely the explicit schemes with a three point stencil, the energy method can be applied for proving stability estimates when the scheme is implemented with either the first or second order extrapolation boundary condition. We then examine the case of five point stencils and give several examples of schemes and second order extrapolation numerical boundary conditions for which the energy method produces stability estimates. However, we also show that for the standard first or second order translatory extrapolation boundary conditions, the energy method cannot be applied for proving stability of the classical fourth order scheme originally proposed by Strang. This gives a clear limitation of the energy method with respect to the more general approach based on the normal mode decomposition.

Published
2020
Full Text
View/download PDF

34. A stiffly stable fully discrete scheme for the damped wave equation using discrete transparent boundary condition

Author

Boutin, Benjamin, Nguyen, Thi Hoai Thuong, Seguin, Nicolas, Boutin, Benjamin, Frontières numériques et couplages - - Nabuco2017 - ANR-17-CE40-0025 - AAPG2017 - VALID, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), and Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)

Subjects
35F46, 35L50, 65M06, 65M12, Z−transform, central schemes, stiff stability, energy estimates, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], hyperbolic relaxation system, damped wave equation, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], discrete transparent boundary condition, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], implicit scheme
Abstract

We study the stability analysis of the time-implicit central differencing scheme for the linear damped wave equation with boundary. Xin and Xu (2000) prove that the initial-boundary value problem (IBVP) for this model is well-posed, uniformly with respect to the stiffness of the damping, under the so-called stiff Kreiss condition (SKC) on the boundary condition. We show here that the (SKC) is also a sufficient condition to guarantee the uniform stability of the discrete IBVP for the relaxation system independently of the stiffness of the source term, of the space step and of the time step. The boundary is approximated using discrete transparent boundary conditions and the stiff stability is proved using energy estimates and the Z−transform.

Published
2020

35. Existence of multi-traveling waves in capillary fluids

Author

Audiard, Corentin, 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 ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Nonlinear Sciences::Exactly Solvable and Integrable Systems, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Nonlinear Sciences::Pattern Formation and Solitons
Abstract

International audience; We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler-Korteweg system in dimension one. Such solutions behaves asymptotically in time like several traveling waves far away from each other. A kink is a traveling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.; On montre l'existence de solutions de type multi-soliton et kink-multi-soliton pour les équations d'Euler-Korteweg en dimension un. Ces solutions se comportent en temps long comme plusieurs ondes solitaires très éloignées les unes des autres. Un kink est une onde solitaire dont les limites en ±∞ sont différentes. L'hypothèse principale est la stabilité linéaire des solitons, on montre que cette hypothèse est satisfaite au moins dans la limite transonique. La preuve suit une approche classique, basée sur des estimées d'énergie et un argument de compacité.

Published
2020

36. Explicit Determination of Robin Parameters in Optimized Schwarz Waveform Relaxation Methods for Schrödinger Equations Based on Pseudodifferential Operators

Author

Xavier Antoine, Emmanuel Lorin, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), School of Mathematics and Statistics [Ottawa], Carleton University, X. Antoine was partially supported by the French National Research Agency project NABUCO, grant ANR-17-CE40-0025. E. Lorin received support from NSERC through the Discovery Grant program., and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Partial differential equation, Physics and Astronomy (miscellaneous), Computational complexity theory, Computer science, Computation, dynamics, Robin boundary condition, 01 natural sciences, Optimized Schwarz Waveform Relaxation, 010101 applied mathematics, Nonlinear system, SchrödingerSchr¨Schrödinger equation, Rate of convergence, Convergence (routing), stationary states, fast convergence, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Decomposition method (constraint satisfaction), 0101 mathematics, pseudodifferential operators, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], domain decomposition method
Abstract

International audience; The Optimized Schwarz Waveform Relaxation algorithm, a domain decomposition method based on Robin transmission condition, is becoming a popular computational method for solving evolution partial differential equations in parallel. Along with well-posedness, it offers a good balance between convergence rate, computational complexity and simplicity of the implementation. The fundamental question is the selection of the Robin parameter to optimize the convergence of the algorithm. In this paper, we propose an approach to explicitly estimate the Robin parameter which is based on the approximation of the transmission operators at the subdomain interfaces, for the linear/nonlinear SchrödingerSchr¨Schrödinger equation. Some illustrating numerical experiments are proposed for the one-and two-dimensional problems.

Published
2020
Full Text
View/download PDF

37. Stability and convergence analysis of artificial boundary conditions for the Schrödinger equation on a rectangular domain

Author

Yibo Yang, Xavier Antoine, Gang Pang, Shaoqiang Tang, Beijing University of Aeronautics and Astronautics (BUAA), University of Pennsylvania [Philadelphia], Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL), Peking University [Beijing], G. PANG and S. TANG are supported partially by NSFC under contract numbers 11502028 and11832001., X. ANTOINE thanks the support of the CNRS LIASFMA (Laboratoire International Associé Sino-Français en Mathématiques Appliquées), University of Lorraine and of the French NationalResearch Agency project NABUCO, grant ANR-17-CE40-0025. Parts of the work were done while X.ANTOINE was a visiting Professor at Peking University (Department of Mechanics and EngineeringScience, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and University of Pennsylvania

Subjects
Algebra and Number Theory, Applied Mathematics, AMS Subject Classification: 35J10, 65L20, 65L10, 65L12, Schrödinger equation, 010103 numerical & computational mathematics, stability analysis, 01 natural sciences, Stability (probability), Domain (mathematical analysis), convergence analysis, 010101 applied mathematics, Computational Mathematics, symbols.namesake, Rate of convergence, Scheme (mathematics), Convergence (routing), symbols, artificial boundary condition, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience; Based on the semi-discrete artificial boundary condition introduced in [24] for the twodimensional free Schrödinger equation in a computational rectangular domain, we propose to analyze the stability and convergence rate with respect to time of the resulting full scheme. We prove that the global scheme is L2-stable and that the accuracy is second-order in time, confirming then the numerical results reported in "Ji S., Yang Y., Pang G., Antoine X., Computer Physics Communications, 2018" [24].

Published
2019
Full Text
View/download PDF

38. On the time of existence of solutions of the Euler-Korteweg system

Author

Audiard, Corentin, Laboratoire Jacques-Louis Lions (LJLL (UMR_7598)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Sorbonne Université (SU), Centre National de la Recherche Scientifique (CNRS), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Audiard, Corentin, and Frontières numériques et couplages - - Nabuco2017 - ANR-17-CE40-0025 - AAPG2017 - VALID

Subjects
Mathematics - Analysis of PDEs, transport, FOS: Mathematics, Mathematics::Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], dispersion, long time existence, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], Euler-Korteweg system, Analysis of PDEs (math.AP)
Abstract

Under a natural stability condition on the pressure, it is known that for small irrotational initial data, the solutions of the Euler-Korteweg system are global in time. When the initial velocity has a small rotational part, we obtain a lower bound on the time of existence that depends only on the rotational part. In the zero vorticity limit we recover the previous global well-posedness result. Independently of this analysis, we also provide (in a special case) a simple example of solution that blows up in finite time., Sous une condition naturelle de stabilité portant sur la pression et en dimension d'espace au moins trois, il est connu que le système d'Euler-Korteweg admet des solutions globales à petites données. Lorsque la vitesse initiale a une composante "rotationnelle" petite mais non nulle, nous obtenons une borne inférieure sur le temps d'existence qui ne dépend que de cette composante. Dans la limite irrotationnelle, nous retrouvons le précédent résultat d'existence globale. Indépendamment de cette analyse, un exemple simple de solution qui explose en temps fini est inclus.

Published
2019

39. Energy-stable staggered schemes for the Shallow Water equations

Author

DURAN, Arnaud, Vila, Jean-Paul, Baraille, Rémy, Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), 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), Institut National des Sciences Appliquées (INSA), Service Hydrographique et Océanographique de la Marine (SHOM), Ministère de la Défense, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), Institut Camille Jordan (ICJ), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet - Saint-Étienne (UJM)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)

Subjects
energy dissipation, non-linear stability, well-balanced methods, non-linear shallow water equations, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], staggered meshes, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; In this work we focus on the development and analysis of staggered schemes for the two-dimensional non-linear Shallow Water equations with varying bathymetry. Semi-implicit and fully explicit time-discretizations are proposed. Particular attention is given on non-linear stability results, principally considered here through discrete energy dissipation arguments. To address such an issue, specific convective fluxes are employed, implying diffusive terms relying on the pressure gradient. In addition of providing an explicit control of the discrete energy budget, the method is shown to preserve motionless steady states as well as the positivity of the water height. These properties are still satisfied in a fully explicit context, provided an appropriate discretization of the pressure gradient. These stability results make the approach particularly robust and efficient, for both coastal flows and low Froude number regimes. As a result, in addition of a great ease of implementation, the presented schemes meet the operational requirements attached to the simulation of large and small scale oceanic flows.

Published
2019

40. Hyperbolic free boundary problems and applications to wave-structure interactions

Author

Tatsuo Iguchi, David Lannes, Department of Mathematics (KEIO UNIVERSITY), 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), Fondation Del Duca, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), and ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018)

Subjects
Trace (linear algebra), General Mathematics, 010102 general mathematics, Mathematical analysis, Boundary (topology), Kinematics, Type (model theory), 01 natural sciences, Stability (probability), Nonlinear system, Mathematics - Analysis of PDEs, FOS: Mathematics, Free boundary problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, 0101 mathematics, Analysis of PDEs (math.AP), Mathematics
Abstract

International audience; Motivated by a new kind of initial boundary value problem (IBVP) with a free boundary arising in wave-structure interaction, we propose here a general approach to one-dimensional IBVP as well as transmission problems. For general strictly hyperbolic 2 × 2 quasilinear hyperbolic systems, we derive new sharp linear estimates with refined dependence on the source term and control on the traces of the solution at the boundary. These new estimates are used to obtain sharp results for quasilinear IBVP and transmission problems, and for fixed, moving, and free boundaries. In the latter case, two kinds of evolution equations are considered. The first one is of " kinematic type " in the sense that the velocity of the interface has the same regularity as the trace of the solution. Several applications that fall into this category are considered: the interaction of waves with a lateral piston, and a new version of the well-known stability of shocks (classical and undercompressive) that improves the results of the general theory by taking advantage of the specificities of the one-dimensional case. We also consider " fully nonlinear " evolution equations characterized by the fact that the velocity of the interface is one derivative more singular than the trace of the solution. This configuration is the most challenging; it is motivated by a free boundary problem arising in wave-structure interaction, namely, the evolution of the contact line between a floating object and the water. This problem is solved as an application of the general theory developed here.

Published
2019

41. Generating boundary conditions for a Boussinesq system

Author

David Lannes, Lisl Weynans, Modélisation et calculs pour l'électrophysiologie cardiaque (CARMEN), IHU-LIRYC, Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-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 Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), 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), Fondation Del Duca, INRIA Bordeaux, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), 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)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-IHU-LIRYC, Université Bordeaux Segalen - Bordeaux 2-CHU Bordeaux [Bordeaux]-CHU Bordeaux [Bordeaux], 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
Finite volumes method, General Physics and Astronomy, Boundary (topology), 01 natural sciences, [SPI.MECA.MEFL]Engineering Sciences [physics]/Mechanics [physics.med-ph]/Fluids mechanics [physics.class-ph], Boussinesq system, Mathematics - Analysis of PDEs, Simple (abstract algebra), FOS: Mathematics, Periodic boundary conditions, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Mathematics - Numerical Analysis, 0101 mathematics, Condition aux limites génératrice, Mathematical Physics, Système de Boussinesq, Mathematics, [PHYS.PHYS.PHYS-AO-PH]Physics [physics]/Physics [physics]/Atmospheric and Oceanic Physics [physics.ao-ph], Finite volume method, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Generating boundary condition, Couche limite dispersive, Statistical and Nonlinear Physics, Numerical Analysis (math.NA), Dispersive boundary layer, 010101 applied mathematics, Boundary layer, Méthode de Volumes Finis, Non homogeneous, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Resolution (algebra), Analysis of PDEs (math.AP)
Abstract

International audience; We present a new method for the numerical implementation of generating boundary conditions for a one dimensional Boussinesq system. This method is based on a reformulation of the equations and a resolution of the dispersive boundary layer that is created at the boundary when the boundary conditions are non homogeneous. This method is implemented for a simple first order finite volume scheme and validated by several numerical simulations. Contrary to the other techniques that can be found in the literature, our approach does not cause any increase in computational time with respect to periodic boundary conditions; Nous présentons une nouvelle méthode pour l’implantation numérique de conditions aux limites génératrices pour un système de Boussinesq en une dimension. Cette méthode est basée sur une reformulation des équations et la résolution d’une couche limite dispersive qui est créée sur la frontièredu domaine quand les conditions aux limites ne sont pas homogènes. Cette méthode est implantée pour un schéma simple de type Volumes Finis d’ordre un et validée par plusieurs simulations numériques. Contrairement aux autres techniques de la littérature, notre approche n’occasionne aucune augmentation de temps de calcul par rapport à des conditions aux limites périodiques

Published
2019
Full Text
View/download PDF

42. Asymptotic convergence rates of SWR methods for Schrödinger equations with an arbitrary number of subdomains

Author

Antoine, Xavier, Lorin, Emmanuel, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), School of Mathematics and Statistics [Ottawa], Carleton University, Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), X. Antoine was supported by the French National Research Agency project NABUCO, Grant ANR-17-CE40-0025., E. Lorin thanks NSERC for the financial support via the Discovery Grant program., and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Schwarz Waveform Relaxation, Mathematics::Complex Variables, asymptotic convergence rate, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Schrödinger equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], domain decomposition method
Abstract

International audience; We derive some estimates of the rate of convergence of Schwarz Waveform Relaxation (SWR) methods for the Schrödinger equation using an arbitrary number of subdomains. Hence, we justify that under certain conditions, the rates of convergence mathematically obtained for two subdomains [6, 7, 8] are still asymptotically valid for a larger number of subdomains, as it is usually numerically observed [22].

Published
2019
Full Text
View/download PDF

43. A simple pseudospectral method for the computation of the time-dependent Dirac equation with Perfectly Matched Layers

Author

Xavier Antoine, Emmanuel Lorin, Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), School of Mathematics and Statistics [Ottawa], Carleton University, Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Physics and Astronomy (miscellaneous), high-order accuracy, Computation, Fast Fourier transform, Boundary (topology), 010103 numerical & computational mathematics, 01 natural sciences, symbols.namesake, Simple (abstract algebra), Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Pseudo-spectral method, Boundary value problem, Dirac equation, 0101 mathematics, pseudospectral approximation, Mathematics, Numerical Analysis, Perfectly Matched Layers, Applied Mathematics, Computer Science Applications, time-splitting, 010101 applied mathematics, Computational Mathematics, Modeling and Simulation, symbols, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

International audience; A simple time-splitting pseudospectral method for the computation of the Dirac equation with Perfectly Matched Layers is proposed. Within this approach, basic and widely used FFT-based solvers can be adapted without much effort to compute Initial Boundary Value Problems for the time-dependent Dirac equation with absorbing boundary layers. Some numerical examples from laser-physics are proposed to illustrate the method.

Published
2019
Full Text
View/download PDF

44. Field data-based evaluation of methods for recovering surface wave elevation from pressure measurements

Author

Arthur Mouragues, Vincent Marieu, David Lannes, Bruno Castelle, Philippe Bonneton, Environnements et Paléoenvironnements OCéaniques (EPOC), Observatoire aquitain des sciences de l'univers (OASU), Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), 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), Fondation Del Duca, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), and Bonneton, Philippe

Subjects
Physics, [SDE] Environmental Sciences, Environmental Engineering, 010504 meteorology & atmospheric sciences, 010505 oceanography, Elevation, Ocean Engineering, Mechanics, Surf zone, 01 natural sciences, Cutoff frequency, Airy wave theory, Surface wave, [SDE]Environmental Sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Crest, Rogue wave, Significant wave height, ComputingMilieux_MISCELLANEOUS, 0105 earth and related environmental sciences
Abstract

We compare different methods to reconstruct the surface elevation of irregular waves propagating outside the surf zone from pressure measurements at the bottom. The traditional transfer function method (TFM), based on the linear wave theory, predicts reasonably well the significant wave height but cannot describe the highest frequencies of the wave spectrum. This is why the TFM cannot reproduce the skewed shape of nonlinear waves and strongly underestimates their crest elevation. The surface elevation reconstructed from the TFM is very sensitive to the value of the cutoff frequency. At the individual wave scale, high-frequency tail correction strategies associated with this method do not significantly improve the prediction of the highest waves. Unlike the TFM, the recently developed weakly-dispersive nonlinear reconstruction method correctly reproduces the wave energy over a large number of harmonics leading to an accurate estimation of the peaked and skewed shape of the highest waves. This method is able to recover the most nonlinear waves within wave groups which some can be characterized as extreme waves. It is anticipated that using relevant reconstruction method will improve the description of individual wave transformation close to breaking.

Published
2019
Full Text
View/download PDF

45. Ghost solutions with centered schemes for one-dimensional transport equations with Neumann boundary conditions

Author

Hans Henrik Rugh, Mélanie Inglard, Frédéric Lagoutière, Modélisation mathématique, calcul scientifique (MMCS), Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
010101 applied mathematics, Mathematical analysis, Neumann boundary condition, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010103 numerical & computational mathematics, General Medicine, 0101 mathematics, 01 natural sciences, ComputingMilieux_MISCELLANEOUS, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract

International audience

Published
2019

46. Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

Author

Jean-Claude Saut, Benoit Desjardins, David Lannes, Modélisation, Mesures et Applications S.A. (MOMA), MOMA, 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 d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Fondation del Duca, Conseil Régional Aquitaine, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), and ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018)

Subjects
Stratification (water), FOS: Physical sciences, Physics - Classical Physics, 01 natural sciences, 010305 fluids & plasmas, Mathematics - Analysis of PDEs, Normal mode, 0103 physical sciences, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [PHYS.MECA.MEFL]Physics [physics]/Mechanics [physics]/Fluid mechanics [physics.class-ph], 0101 mathematics, Boussinesq approximation (water waves), [SDU.OCEAN]Sciences of the Universe [physics]/Ocean, Atmosphere, Physics, Partial differential equation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Classical Physics (physics.class-ph), Internal wave, Wave equation, Computational Mathematics, Nonlinear system, Modeling and Simulation, Linear approximation, Analysis, Analysis of PDEs (math.AP)
Abstract

International audience; Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical strati-fication of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm-Liouville problem associated to the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so called Brunt-Vaisälä frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approximation (i.e. neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a non-linear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluids systems.

Published
2019
Full Text
View/download PDF

47. A simple pseudospectral method for the computation of the time-dependent Dirac equation with Perfectly Matched Layers. Application to quantum relativistic laser physics

Author

Antoine, X, Lorin, E, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization (SPHINX), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Centre de Recherches Mathématiques [Montréal] (CRM), Université de Montréal (UdeM), School of Mathematics and Statistics [Ottawa], Carleton University, X. ANTOINE was partially supported by the French National Re-search Agency project NABUCO, grant ANR-17-CE40-0025., E. LORIN received supportfrom NSERC through the Discovery Grant program., and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
Perfectly Matched Layers, high-order accuracy, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Dirac equation, pseudospectral approximation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], time-splitting
Abstract

A simple time-splitting pseudospectral method for the computation of the Dirac equation with Perfectly Matched Layers (PML) is proposed. Within this approach, basic and widely used FFT-based solvers can be adapted without much effort to compute Initial Boundary Value Problems (IBVP) for the time-dependent Dirac equation with absorbing boundary layers. Some numerical examples from laser-physics are proposed to illustrate the method.

Published
2018

48. UHAINA : A parallel high performance unstructured near-shore wave model

Author

Filippini, Andrea Gilberto, De Brye, Sébastien, Perrier, Vincent, Marche, Fabien, Ricchiuto, Mario, Lannes, David, Bonneton, Philippe, Certified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts (CARDAMOM), 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 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)-Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Environnements et Paléoenvironnements OCéaniques (EPOC), Observatoire aquitain des sciences de l'univers (OASU), Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Computational AGility for internal flows sImulations and compaRisons with Experiments (CAGIRE), Inria Bordeaux - Sud-Ouest, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Pau et des Pays de l'Adour (UPPA), Laboratoire de Mathématiques et de leurs Applications [Pau] (LMAP), Université de Pau et des Pays de l'Adour (UPPA)-Centre National de la Recherche Scientifique (CNRS), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-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), Daniel Levacher, Martin Sanchez, Xavier Bertin, Isabelle Brenon, ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), ANR-18-CE40-0027,SingFlows,Ecoulements avec singularités : couches limites, filaments de vortex, interaction vague-structure(2018), 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)-Inria Bordeaux - Sud-Ouest, Université Sciences et Technologies - Bordeaux 1 (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1 (UB)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École Pratique des Hautes Études (EPHE), 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
Wave breaking, Discontinuous Finite Element, Phase-resolving wave model, Unstructured meshes, Green Naghdi equations, [INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation
Abstract

International audience; UHAINA is a new phase-resolving free surface wave model for coastal engineering 21 problems. It is based on the most advanced and recent contributions in coastal 22 modelling from the french institutes EPOC, IMAG, IMB, and INRIA BSO. It solves a 23 non-classical version of the depth-integrated fully-nonlinear and weakly-dispersive 24 equations of Green-Naghdi, which allows an efficient numerical implementation. 25 UHAINA relies on libraries developed at the INRIA BSO center, such as AeroSol for 26 its hydrodynamic core, and PaMPA and SCOTCH to handle data management for 27 distributed memory parallel computation. The use of these libraries, in particular 28 AeroSol, offers a wide range of possibilities including arbitrary high-order finite 29 element discretizations, hybrid meshes (structured and unstructured), as well as an 30 advanced programming environment specially designed by the purpose of performance 31 and HPC. These properties will lead in the coming years to the release of a new efficient 32 and robust open source wave modelling platform, available for a large community of 33 users and very suitable for practical coastal applications.

Published
2018
Full Text
View/download PDF

49. Étude comparative des méthodes de reconstruction du champ de vagues à partir de la mesure de pression : Application à la plage de La Salie

Author

Arthur Mouragues, David Lannes, Guillaume Detandt, Benjamin Dubarbier, Pierre-Antoine Poncet, Bruno Castelle, Philippe Bonneton, Vincent Marieu, Environnements et Paléoenvironnements OCéaniques (EPOC), Observatoire aquitain des sciences de l'univers (OASU), Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Sciences et Technologies - Bordeaux 1-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-École pratique des hautes études (EPHE), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), UMR 5805 Environnements et Paléoenvironnements Océaniques et Continentaux (EPOC), ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017), and École normale supérieure - Paris (ENS Paris)

Subjects
[SDE]Environmental Sciences
Abstract

International audience; La reconstruction du champ de vagues basée sur la théorie linéaire, communément appelée méthode de la fonction de transfert, décrit assez bien les caractéristiques moyennes des vagues. Elle est couramment utilisée pour déterminer les paramètres statistiques des vagues mais sous-estime la hauteur des plus grosses vagues. Cependant, bien estimer ces vagues énergétiques est primordial pour l'étude des risques côtiers lors d'évènements extrêmes. La reconstruction linéaire nécessite l'application d'une fréquence de coupure du spectre d'énergie qui induit une perte d'information. Des techniques empiriques qui essaient de combler artificiellement le spectre d'énergie n'améliorent que très peu les plus grosses vagues. Récemment, des formules simples à implémenter permettant des reconstructions linéaires et non-linéaires ont été développées. Nous présentons et comparons dans cet article différentes méthodes de reconstruction du champ de vagues à partir de la pression mesurée au fond lors d'une campagne de mesure réalisée sur la plage de La Salie en avril 2017. Notre étude est centrée sur des vagues très non-linéaires et faiblement dispersives. Une analyse vague à vague met en évidence la capacité de la reconstruction non-linéaire faiblement dispersive à reproduire la hauteur et l'asymétrie des plus grosses vagues contrairement à la formule linéaire couramment utilisée en ingénierie côtière. Mots-clés : Théorie linéaire, non-linéarité, capteurs de pression, reconstruction du champ de vagues.

Published
2018
Full Text
View/download PDF

50. Several models for wave-structure interactions

Author

Lannes, David, 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), Fondation Del Duca, Conseil Régional d'Aquitaine, and ANR-17-CE40-0025,Nabuco,Frontières numériques et couplages(2017)

Subjects
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ComputingMilieux_MISCELLANEOUS
Abstract

International audience

Published
2018

Catalog

Books, media, physical & digital resources
See catalog results