This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes... more This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes when approximating the weak solutions of the shallow water model. More precisely, here we adopt the well-known hydrostatic reconstruction technique to enforce the adopted scheme to be well-balanced; namely to exactly preserve the lake at rest stationary solution. Such a numerical approach is known to get a semi-discrete (continuous in time) entropy inequality. However, a semi-discrete energy estimation turns, in general, to be insufficient to claim the required stability. In the present work, we adopt the artificial numerical viscosity technique to increase the desired stability and then to recover a fully discrete energy estimate. Several numerical experiments illustrate the relevance of the designed viscous hydrostatic reconstruction scheme.
In this paper, we propose a discretization of the multi-dimensional stationary compressible Navie... more In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p(ρ) = aρ , with γ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents γ less than 3 when the space dimension is three. The present convergence result can be seen as a discrete counterpart of the construction of weak solutions established by P.-L. Lions and by S. Novo, A. Novotný.
We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Ba... more We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's Godunov-type scheme [39] and Toro-Tokareva's HLLC scheme [42]. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.
We study a model for compressible multiphase flows involving N non miscible barotopic phases wher... more We study a model for compressible multiphase flows involving N non miscible barotopic phases where N is arbitrary. This model boils down to the barotropic Baer-Nunziato model when N = 2. We prove the weak hyperbolicity property, the non-strict convexity of the natural mathematical entropy, and the existence of a symmetric form.
In this paper we analyze two numerical staggered schemes, one fully implicit, the other semiimpli... more In this paper we analyze two numerical staggered schemes, one fully implicit, the other semiimplicit, for the compressible Navier-Stokes equations with a singular pressure law; system which degenerates towards the free-congested Navier-Stokes equations with the density constraint 0 ≤ ρ ≤ 1 as the intensity of the pressure, ε, tends to 0. The two schemes are shown to "preserve" the limit ε → 0, in the following sense: via discrete energy and pressure estimates we control uniformly with respect to ε the discrete density, velocity and pressure; and, as ε → 0, the limit pressure is shown to be activated only in the congested region where the discrete limit density is maximal, equal to 1. Finally, the semi-implicit scheme is implemented via the software CALIF 3 S on 1D and 2D test cases and the numerical solutions are shown to capture well the transition between the congested regions {ρ = 1} and the free regions {ρ < 1}.
In this paper, we study the behaviour of some staggered discretization based numerical schemes fo... more In this paper, we study the behaviour of some staggered discretization based numerical schemes for the barotropic Navier-Stokes equations at low Mach number. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The two latter schemes differ by the discretization of the convection term: linearly implicit for the first one, so that the resulting scheme is unconditionally stable, and explicit for the second one, so the scheme is stable under a CFL condition involving the material velocity only. We rigorously prove that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tend to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof of convergence of the solutions of the (continuous) barotropic Navier-Stokes equations to that of the incompressible Navier-Stokes equation as the Mach number vanishes. Numerical results performed with a hand-built analytical solution show the behaviour that is expected from the analysis.
qui je souhaite une excellente fin de thèse sur Baer-Nunziato !), El Hassan, Mohamed Abaidi (« po... more qui je souhaite une excellente fin de thèse sur Baer-Nunziato !), El Hassan, Mohamed Abaidi (« pourquoi, y'a deux Célines ? ») et Mohamed Ghattassi. Merci aussi à Nina, Mathieu, Rémi, David et Fabien, pour n'avoir jamais voulu m'apprendre la Coinche... Merci enfin aux strasbourgeois, Anaïs, Hélène, et les garçons : Jonathan et Ahmed. Ahmed, qui m'a appris, à moi l'Egyptien, comment préparer une chicha. Merci à Mohamed Jaoua de m'avoir permis d'enseigner à l'Université Française d'Egypte. Merci aussi à Manuel pour ses excellents plans resto au Caire ! Avant de commencer cette thèse, j'ai eu le plaisir de suivre les cours du master ANEDP de Paris 6, où j'ai fait la connaissance de jeunes chercheurs d'horizons très différents, devenus pour certains des amis. Je pense à Nafi, Maxime, Cécile, Manu, Morgan et Aurore, et je souhaite bon courage à ceux d'entre eux qui finissent bientôt leur thèse. Je remercie également mes meilleurs amis, Wassef, Ali et Sullym (dans l'ordre chronologique de rencontre). Je crois que je ne compte plus les fous rires qu'on a pu avoir ensemble ! Merci à vous trois d'avoir été là et de m'avoir permis, chacun à sa façon, de sortir un petit peu de mon univers de matheux. Un grand MERCI à ma famille, en France et en Egypte, pour m'avoir soutenu durant toutes mes années d'études. Merci à mes frères Nouredine et Nassim (les lascars, experts de la vanne) d'avoir manqué vos cours pour assister à ma soutenance. Bon courage pour vos études, et si jamais vous avez des questions, on va voir ce qu'on peut faire. Merci à ma soeur Ghadah ainsi qu'à son époux Mohamed, et bienvenue à mon petit neveu Adam (plus tard, tu pourras dire que j'ai parlé de toi dans ma thèse). Et surtout, surtout, merci à mes parents, que j'aime tendrement. Merci pour tous les sacrifices que vous avez faits et continuez de faire pour nous. Que Dieu vous garde à nos côtés. Enfin, je dédie ce travail à ma très chère femme Haïfa. Merci pour ton amour, ta tendresse, ton amitié, ton soutien dans les moments difficiles. Toi qui as toujours su m'encourager et me donner confiance en moi, merci d'être là. Tout ceci n'est rien sans toi.
Springer Proceedings in Mathematics & Statistics, 2017
We study the incompressible limit of a pressure correction MAC scheme (Herbin et al., Math. Model... more We study the incompressible limit of a pressure correction MAC scheme (Herbin et al., Math. Model. Numer. Anal. 48, 1807-1857, 2013) [3] for the unstationary compressible barotropic Navier-Stokes equations. Provided the initial data are well-prepared, the solution of the numerical scheme converges, as the Mach number tends to zero, towards the solution of the classical pressure correction inf-sup stable MAC scheme for the incompressible Navier-Stokes equations. Keywords Compressible Navier-Stokes equations • Low Mach number flows • Finite volumes • MAC scheme • Staggered discretizations MSC (2010) 35Q30 • 65N12 • 76M12 1 Introduction Let Ω be a parallelepiped of R d , with d ∈ {2, 3} and T > 0. The unsteady barotropic compressible Navier-Stokes equations, parametrized by the Mach number ε, read for (x, t
When considering the numerical approximation of weak solutions of systems of conservation laws, t... more When considering the numerical approximation of weak solutions of systems of conservation laws, the derivation of discrete entropy inequalities is, in general, very difficult to obtain. In the present work, we present a suitable control of the numerical artificial viscosity in order to recover the expected discrete entropy inequalities. Moreover, the artificial viscosity control turns out to be very easy and the resulting numerical implementation is very convenient.
In this paper, we propose a discretization of the multi-dimensional stationary compressible Navie... more In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p($\rho$) = a$\rho$ $\gamma$ , with $\gamma$ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents $\gamma$ less than 3 when the space dimension is three. The present convergence result can be seen as a discrete counterpart of the construction of weak solutions established by P.-L. Lions and by S. Novo, A. Novotn{ý}.
Cette these s'interesse au modele diphasique de Baer-Nunziato. L'objectif de ce travail e... more Cette these s'interesse au modele diphasique de Baer-Nunziato. L'objectif de ce travail est de proposer quelques techniques de prise en compte de la disparition de phase, regime occasionnant d'importantes instabilites au niveau du modele et de sa simulation numerique. Par des methodes d'analyse et de simulation reposant sur les techniques d'approximation par relaxation a la Suliciu, on montre que dans ces regimes, on peut stabiliser les solutions en introduisant une dissipation de l'entropie totale de melange. Dans une premiere approche dite approche Eulerienne directe, la resolution exacte du probleme de Riemann pour le systeme relaxe permet de definir un schema entropique extremement precis, et qui se revele bien plus economique en terme de cout CPU (a precision donnee) que le schema classique tres simple de Rusanov. De plus, nous montrons que ce schema permet de simuler avec robustesse des regimes de disparition de phase. Le schema est developpe en 1D puis...
In this paper, we build and analyze a scheme for the time-dependent variable density Navier-Stoke... more In this paper, we build and analyze a scheme for the time-dependent variable density Navier-Stokes equations, able to cope with unstructured non-conforming meshes, with hanging nodes. The time advancement relies on a pressure correction algorithm and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, and a careful construction of the fluxes, especially through non-conforming faces, ensures that the solution obeys a discrete kinetic energy balance. Its consistency is addressed by analyzing a model problem, namely the convection-diffusion equation, for which we theoretically establish a first order convergence in space for energy norms. This convergence order is also observed in the numerical experiments for the Navier-Stokes equations. Navier-Stokes equations, pressure correction scheme, finite volumes, finite e...
In this paper we prove a convergence result for a discretization of the three-dimensional station... more In this paper we prove a convergence result for a discretization of the three-dimensional stationary compressible Navier–Stokes equations assuming an ideal gas pressure law $p(\rho )=a \rho ^{\gamma }$ with $\gamma> \frac{3}{2}$. It is the first convergence result for a numerical method with adiabatic exponents $\gamma $ less than $3$ when the space dimension is 3. The considered numerical scheme combines finite volume techniques for the convection with the Crouzeix–Raviart finite element for the diffusion. A linearized version of the scheme is implemented in the industrial software CALIF3S developed by the French Institut de Radioprotection et de Sûreté Nucléaire.
In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equation... more In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem (L ∞-estimate for the density, L ∞ (L 2)-and L 2 (H 1)-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.
We present here a Godunov-type scheme to simulate one-dimensional flows in a nozzle with variable... more We present here a Godunov-type scheme to simulate one-dimensional flows in a nozzle with variable cross-section. The method relies on the construction of a relaxation Riemann solver designed to handle all types of flow regimes, from subsonic to supersonic flows, as well as resonant transonic flows. Some computational results are also provided, in which this relaxation method is compared with the classical Rusanov scheme and a modified Rusanov scheme.
42nd AIAA Fluid Dynamics Conference and Exhibit, 2012
We introduce a class of two-fluid models that complies with a few theoretical requirements that i... more We introduce a class of two-fluid models that complies with a few theoretical requirements that include : (i) hyperbolicity of the convective subset, (ii) entropy inequality, (iii) uniqueness of jump conditions for non-viscous flows. These specifications are necessary in order to compute relevant approximations of unsteady flow patterns. It is shown that the Baer-Nunziato model belongs to this class of two-phase flow models, and the main properties of the model are given, before showing a few numerical experiments.
We propose in this work to address the problem of model adaptation, dedicated to hyperbolic model... more We propose in this work to address the problem of model adaptation, dedicated to hyperbolic models with relaxation and to their parabolic limit. The goal is to replace a hyperbolic system of balance laws (the so-called fine model) by its parabolic limit (the so-called coarse model), in delimited parts of the computational domain. Our method is based on the construction of asymptotic preserving schemes and on interfacial coupling methods between hyperbolic and parabolic models. We study in parallel the cases of the Goldstein-Taylor model and of the p-system with friction. Résumé. Nous proposons dans ce travail de traiter le problème d'adaptation de modèle, appliqué aux systèmes hyperboliques de relaxation età leur limite parabolique. Le but est de remplacer dans des zones délimitées du domaine de calcul un système hyperbolique avec terme source (le modèle fin) par le modèle parabolique limite associé (le modèle grossier). Notre méthode repose sur des schémas préservant cette asymptotique et le couplage interfacial entre des modèles hyperbolique et parabolique. Onétudie les cas du modèle de Goldstein-Taylor et du p-système avec friction avec leurs limites paraboliques respectives. Contents * This project has been supported by the LRC Manon (Modélisation et Approximation Numérique Orientées pour l'énergie Nucléaire-CEA/DM2S-LJLL).
ESAIM: Mathematical Modelling and Numerical Analysis, 2013
We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model,... more We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.
ESAIM: Mathematical Modelling and Numerical Analysis, 2019
This article is the first of two in which we develop a relaxation finite volume scheme for the co... more This article is the first of two in which we develop a relaxation finite volume scheme for the convective part of the multiphase flow models introduced in the series of papers \cite{her-16-cla,Herard,bou-18-rel}. In the present article we focus on barotropic flows where in each phase the pressure is a given function of the density. The case of general equations of state will be the purpose of the second article. We show how it is possible to extend the relaxation scheme designed in \cite{coq-13-rob} for the barotropic Baer-Nunziato two phase flow model to the multiphase flow model with $N$ - where $N$ is arbitrarily large - phases. The obtained scheme inherits the main properties of the relaxation scheme designed for the Baer-Nunziato two phase flow model. It applies to general barotropic equations of state. It is able to cope with arbitrarily small values of the statistical phase fractions. The approximated phase fractions and phase densities are proven to remain positive and a ful...
This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes... more This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes when approximating the weak solutions of the shallow water model. More precisely, here we adopt the well-known hydrostatic reconstruction technique to enforce the adopted scheme to be well-balanced; namely to exactly preserve the lake at rest stationary solution. Such a numerical approach is known to get a semi-discrete (continuous in time) entropy inequality. However, a semi-discrete energy estimation turns, in general, to be insufficient to claim the required stability. In the present work, we adopt the artificial numerical viscosity technique to increase the desired stability and then to recover a fully discrete energy estimate. Several numerical experiments illustrate the relevance of the designed viscous hydrostatic reconstruction scheme.
In this paper, we propose a discretization of the multi-dimensional stationary compressible Navie... more In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p(ρ) = aρ , with γ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents γ less than 3 when the space dimension is three. The present convergence result can be seen as a discrete counterpart of the construction of weak solutions established by P.-L. Lions and by S. Novo, A. Novotný.
We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Ba... more We present a relaxation scheme for approximating the entropy dissipating weak solutions of the Baer-Nunziato two-phase flow model. This relaxation scheme is straightforwardly obtained as an extension of the relaxation scheme designed in [16] for the isentropic Baer-Nunziato model and consequently inherits its main properties. To our knowledge, this is the only existing scheme for which the approximated phase fractions, phase densities and phase internal energies are proven to remain positive without any restrictive condition other than a classical fully computable CFL condition. For ideal gas and stiffened gas equations of state, real values of the phasic speeds of sound are also proven to be maintained by the numerical scheme. It is also the only scheme for which a discrete entropy inequality is proven, under a CFL condition derived from the natural sub-characteristic condition associated with the relaxation approximation. This last property, which ensures the non-linear stability of the numerical method, is satisfied for any admissible equation of state. We provide a numerical study for the convergence of the approximate solutions towards some exact Riemann solutions. The numerical simulations show that the relaxation scheme compares well with two of the most popular existing schemes available for the Baer-Nunziato model, namely Schwendeman-Wahle-Kapila's Godunov-type scheme [39] and Toro-Tokareva's HLLC scheme [42]. The relaxation scheme also shows a higher precision and a lower computational cost (for comparable accuracy) than a standard numerical scheme used in the nuclear industry, namely Rusanov's scheme. Finally, we assess the good behavior of the scheme when approximating vanishing phase solutions.
We study a model for compressible multiphase flows involving N non miscible barotopic phases wher... more We study a model for compressible multiphase flows involving N non miscible barotopic phases where N is arbitrary. This model boils down to the barotropic Baer-Nunziato model when N = 2. We prove the weak hyperbolicity property, the non-strict convexity of the natural mathematical entropy, and the existence of a symmetric form.
In this paper we analyze two numerical staggered schemes, one fully implicit, the other semiimpli... more In this paper we analyze two numerical staggered schemes, one fully implicit, the other semiimplicit, for the compressible Navier-Stokes equations with a singular pressure law; system which degenerates towards the free-congested Navier-Stokes equations with the density constraint 0 ≤ ρ ≤ 1 as the intensity of the pressure, ε, tends to 0. The two schemes are shown to "preserve" the limit ε → 0, in the following sense: via discrete energy and pressure estimates we control uniformly with respect to ε the discrete density, velocity and pressure; and, as ε → 0, the limit pressure is shown to be activated only in the congested region where the discrete limit density is maximal, equal to 1. Finally, the semi-implicit scheme is implemented via the software CALIF 3 S on 1D and 2D test cases and the numerical solutions are shown to capture well the transition between the congested regions {ρ = 1} and the free regions {ρ < 1}.
In this paper, we study the behaviour of some staggered discretization based numerical schemes fo... more In this paper, we study the behaviour of some staggered discretization based numerical schemes for the barotropic Navier-Stokes equations at low Mach number. Three time discretizations are considered: the implicit-in-time scheme and two non-iterative pressure correction schemes. The two latter schemes differ by the discretization of the convection term: linearly implicit for the first one, so that the resulting scheme is unconditionally stable, and explicit for the second one, so the scheme is stable under a CFL condition involving the material velocity only. We rigorously prove that these three variants are asymptotic preserving in the following sense: for a given mesh and a given time step, a sequence of solutions obtained with a sequence of vanishing Mach numbers tend to a solution of a standard scheme for incompressible flows. This convergence result is obtained by mimicking the proof of convergence of the solutions of the (continuous) barotropic Navier-Stokes equations to that of the incompressible Navier-Stokes equation as the Mach number vanishes. Numerical results performed with a hand-built analytical solution show the behaviour that is expected from the analysis.
qui je souhaite une excellente fin de thèse sur Baer-Nunziato !), El Hassan, Mohamed Abaidi (« po... more qui je souhaite une excellente fin de thèse sur Baer-Nunziato !), El Hassan, Mohamed Abaidi (« pourquoi, y'a deux Célines ? ») et Mohamed Ghattassi. Merci aussi à Nina, Mathieu, Rémi, David et Fabien, pour n'avoir jamais voulu m'apprendre la Coinche... Merci enfin aux strasbourgeois, Anaïs, Hélène, et les garçons : Jonathan et Ahmed. Ahmed, qui m'a appris, à moi l'Egyptien, comment préparer une chicha. Merci à Mohamed Jaoua de m'avoir permis d'enseigner à l'Université Française d'Egypte. Merci aussi à Manuel pour ses excellents plans resto au Caire ! Avant de commencer cette thèse, j'ai eu le plaisir de suivre les cours du master ANEDP de Paris 6, où j'ai fait la connaissance de jeunes chercheurs d'horizons très différents, devenus pour certains des amis. Je pense à Nafi, Maxime, Cécile, Manu, Morgan et Aurore, et je souhaite bon courage à ceux d'entre eux qui finissent bientôt leur thèse. Je remercie également mes meilleurs amis, Wassef, Ali et Sullym (dans l'ordre chronologique de rencontre). Je crois que je ne compte plus les fous rires qu'on a pu avoir ensemble ! Merci à vous trois d'avoir été là et de m'avoir permis, chacun à sa façon, de sortir un petit peu de mon univers de matheux. Un grand MERCI à ma famille, en France et en Egypte, pour m'avoir soutenu durant toutes mes années d'études. Merci à mes frères Nouredine et Nassim (les lascars, experts de la vanne) d'avoir manqué vos cours pour assister à ma soutenance. Bon courage pour vos études, et si jamais vous avez des questions, on va voir ce qu'on peut faire. Merci à ma soeur Ghadah ainsi qu'à son époux Mohamed, et bienvenue à mon petit neveu Adam (plus tard, tu pourras dire que j'ai parlé de toi dans ma thèse). Et surtout, surtout, merci à mes parents, que j'aime tendrement. Merci pour tous les sacrifices que vous avez faits et continuez de faire pour nous. Que Dieu vous garde à nos côtés. Enfin, je dédie ce travail à ma très chère femme Haïfa. Merci pour ton amour, ta tendresse, ton amitié, ton soutien dans les moments difficiles. Toi qui as toujours su m'encourager et me donner confiance en moi, merci d'être là. Tout ceci n'est rien sans toi.
Springer Proceedings in Mathematics & Statistics, 2017
We study the incompressible limit of a pressure correction MAC scheme (Herbin et al., Math. Model... more We study the incompressible limit of a pressure correction MAC scheme (Herbin et al., Math. Model. Numer. Anal. 48, 1807-1857, 2013) [3] for the unstationary compressible barotropic Navier-Stokes equations. Provided the initial data are well-prepared, the solution of the numerical scheme converges, as the Mach number tends to zero, towards the solution of the classical pressure correction inf-sup stable MAC scheme for the incompressible Navier-Stokes equations. Keywords Compressible Navier-Stokes equations • Low Mach number flows • Finite volumes • MAC scheme • Staggered discretizations MSC (2010) 35Q30 • 65N12 • 76M12 1 Introduction Let Ω be a parallelepiped of R d , with d ∈ {2, 3} and T > 0. The unsteady barotropic compressible Navier-Stokes equations, parametrized by the Mach number ε, read for (x, t
When considering the numerical approximation of weak solutions of systems of conservation laws, t... more When considering the numerical approximation of weak solutions of systems of conservation laws, the derivation of discrete entropy inequalities is, in general, very difficult to obtain. In the present work, we present a suitable control of the numerical artificial viscosity in order to recover the expected discrete entropy inequalities. Moreover, the artificial viscosity control turns out to be very easy and the resulting numerical implementation is very convenient.
In this paper, we propose a discretization of the multi-dimensional stationary compressible Navie... more In this paper, we propose a discretization of the multi-dimensional stationary compressible Navier-Stokes equations combining finite element and finite volume techniques. As the mesh size tends to 0, the numerical solutions are shown to converge (up to a subsequence) towards a weak solution of the continuous problem for ideal gas pressure laws p($\rho$) = a$\rho$ $\gamma$ , with $\gamma$ > 3/2 in the three-dimensional case. It is the first convergence result for a numerical method with adiabatic exponents $\gamma$ less than 3 when the space dimension is three. The present convergence result can be seen as a discrete counterpart of the construction of weak solutions established by P.-L. Lions and by S. Novo, A. Novotn{ý}.
Cette these s'interesse au modele diphasique de Baer-Nunziato. L'objectif de ce travail e... more Cette these s'interesse au modele diphasique de Baer-Nunziato. L'objectif de ce travail est de proposer quelques techniques de prise en compte de la disparition de phase, regime occasionnant d'importantes instabilites au niveau du modele et de sa simulation numerique. Par des methodes d'analyse et de simulation reposant sur les techniques d'approximation par relaxation a la Suliciu, on montre que dans ces regimes, on peut stabiliser les solutions en introduisant une dissipation de l'entropie totale de melange. Dans une premiere approche dite approche Eulerienne directe, la resolution exacte du probleme de Riemann pour le systeme relaxe permet de definir un schema entropique extremement precis, et qui se revele bien plus economique en terme de cout CPU (a precision donnee) que le schema classique tres simple de Rusanov. De plus, nous montrons que ce schema permet de simuler avec robustesse des regimes de disparition de phase. Le schema est developpe en 1D puis...
In this paper, we build and analyze a scheme for the time-dependent variable density Navier-Stoke... more In this paper, we build and analyze a scheme for the time-dependent variable density Navier-Stokes equations, able to cope with unstructured non-conforming meshes, with hanging nodes. The time advancement relies on a pressure correction algorithm and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, and a careful construction of the fluxes, especially through non-conforming faces, ensures that the solution obeys a discrete kinetic energy balance. Its consistency is addressed by analyzing a model problem, namely the convection-diffusion equation, for which we theoretically establish a first order convergence in space for energy norms. This convergence order is also observed in the numerical experiments for the Navier-Stokes equations. Navier-Stokes equations, pressure correction scheme, finite volumes, finite e...
In this paper we prove a convergence result for a discretization of the three-dimensional station... more In this paper we prove a convergence result for a discretization of the three-dimensional stationary compressible Navier–Stokes equations assuming an ideal gas pressure law $p(\rho )=a \rho ^{\gamma }$ with $\gamma> \frac{3}{2}$. It is the first convergence result for a numerical method with adiabatic exponents $\gamma $ less than $3$ when the space dimension is 3. The considered numerical scheme combines finite volume techniques for the convection with the Crouzeix–Raviart finite element for the diffusion. A linearized version of the scheme is implemented in the industrial software CALIF3S developed by the French Institut de Radioprotection et de Sûreté Nucléaire.
In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equation... more In this paper, we analyze a scheme for the time-dependent variable density Navier-Stokes equations. The algorithm is implicit in time, and the space approximation is based on a low-order staggered non-conforming finite element, the so-called Rannacher-Turek element. The convection term in the momentum balance equation is discretized by a finite volume technique, in such a way that a solution obeys a discrete kinetic energy balance, and the mass balance is approximated by an upwind finite volume method. We first show that the scheme preserves the stability properties of the continuous problem (L ∞-estimate for the density, L ∞ (L 2)-and L 2 (H 1)-estimates for the velocity), which yields, by a topological degree technique, the existence of a solution. Then, invoking compactness arguments and passing to the limit in the scheme, we prove that any sequence of solutions (obtained with a sequence of discretizations the space and time step of which tend to zero) converges up to the extraction of a subsequence to a weak solution of the continuous problem.
We present here a Godunov-type scheme to simulate one-dimensional flows in a nozzle with variable... more We present here a Godunov-type scheme to simulate one-dimensional flows in a nozzle with variable cross-section. The method relies on the construction of a relaxation Riemann solver designed to handle all types of flow regimes, from subsonic to supersonic flows, as well as resonant transonic flows. Some computational results are also provided, in which this relaxation method is compared with the classical Rusanov scheme and a modified Rusanov scheme.
42nd AIAA Fluid Dynamics Conference and Exhibit, 2012
We introduce a class of two-fluid models that complies with a few theoretical requirements that i... more We introduce a class of two-fluid models that complies with a few theoretical requirements that include : (i) hyperbolicity of the convective subset, (ii) entropy inequality, (iii) uniqueness of jump conditions for non-viscous flows. These specifications are necessary in order to compute relevant approximations of unsteady flow patterns. It is shown that the Baer-Nunziato model belongs to this class of two-phase flow models, and the main properties of the model are given, before showing a few numerical experiments.
We propose in this work to address the problem of model adaptation, dedicated to hyperbolic model... more We propose in this work to address the problem of model adaptation, dedicated to hyperbolic models with relaxation and to their parabolic limit. The goal is to replace a hyperbolic system of balance laws (the so-called fine model) by its parabolic limit (the so-called coarse model), in delimited parts of the computational domain. Our method is based on the construction of asymptotic preserving schemes and on interfacial coupling methods between hyperbolic and parabolic models. We study in parallel the cases of the Goldstein-Taylor model and of the p-system with friction. Résumé. Nous proposons dans ce travail de traiter le problème d'adaptation de modèle, appliqué aux systèmes hyperboliques de relaxation età leur limite parabolique. Le but est de remplacer dans des zones délimitées du domaine de calcul un système hyperbolique avec terme source (le modèle fin) par le modèle parabolique limite associé (le modèle grossier). Notre méthode repose sur des schémas préservant cette asymptotique et le couplage interfacial entre des modèles hyperbolique et parabolique. Onétudie les cas du modèle de Goldstein-Taylor et du p-système avec friction avec leurs limites paraboliques respectives. Contents * This project has been supported by the LRC Manon (Modélisation et Approximation Numérique Orientées pour l'énergie Nucléaire-CEA/DM2S-LJLL).
ESAIM: Mathematical Modelling and Numerical Analysis, 2013
We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model,... more We construct an approximate Riemann solver for the isentropic Baer−Nunziato two-phase flow model, that is able to cope with arbitrarily small values of the statistical phase fractions. The solver relies on a relaxation approximation of the model for which the Riemann problem is exactly solved for subsonic relative speeds. In an original manner, the Riemann solutions to the linearly degenerate relaxation system are allowed to dissipate the total energy in the vanishing phase regimes, thereby enforcing the robustness and stability of the method in the limits of small phase fractions. The scheme is proved to satisfy a discrete entropy inequality and to preserve positive values of the statistical fractions and densities. The numerical simulations show a much higher precision and a more reduced computational cost (for comparable accuracy) than standard numerical schemes used in the nuclear industry. Finally, two test-cases assess the good behavior of the scheme when approximating vanishing phase solutions.
ESAIM: Mathematical Modelling and Numerical Analysis, 2019
This article is the first of two in which we develop a relaxation finite volume scheme for the co... more This article is the first of two in which we develop a relaxation finite volume scheme for the convective part of the multiphase flow models introduced in the series of papers \cite{her-16-cla,Herard,bou-18-rel}. In the present article we focus on barotropic flows where in each phase the pressure is a given function of the density. The case of general equations of state will be the purpose of the second article. We show how it is possible to extend the relaxation scheme designed in \cite{coq-13-rob} for the barotropic Baer-Nunziato two phase flow model to the multiphase flow model with $N$ - where $N$ is arbitrarily large - phases. The obtained scheme inherits the main properties of the relaxation scheme designed for the Baer-Nunziato two phase flow model. It applies to general barotropic equations of state. It is able to cope with arbitrarily small values of the statistical phase fractions. The approximated phase fractions and phase densities are proven to remain positive and a ful...
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Papers by Khaled SALEH