Publications HAL de Mehmet,Ersoy

Journal articles

2021

auteur
Jean-Baptiste Clément, Frederic Golay, Mehmet Ersoy, Damien Sous
titre
An adaptive strategy for discontinuous Galerkin simulations of Richards’ equation: application to multi-materials dam wetting
article
resume
Numerical solution of Richards’ equation remains challenging to get robust, accurate and cost-effective results, particularly for moving sharp wetting fronts. An adaptive strategy for both space and time is proposed to deal with 2D sharp wetting fronts associated with varying and possibly vanishing diffusivity caused by nonlinearity, heterogeneity and anisotropy. Adaptive time stepping makes nonlinear convergence reliable and backward difference formula provides high-order time scheme. Adaptive mesh refinement tracks wetting fronts with an a posteriori error indicator. The novelty of this paper consists in using this technique in combination with a weighted discontinuous Galerkin framework to better approximate steep wetting fronts by a discontinuity. The potential of the overall approach is shown through various examples including analytical and laboratory benchmarks and simulation of full-scale multi-materials dam wetting experiment.
DOI
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2020

auteur
Mehmet Ersoy, Omar Lakkis, Philip Townsend
titre
A Saint-Venant Model for Overland Flows with Precipitation and Recharge
article
Mathematical and computational applications, 2020, ⟨10.3390/mca26010001⟩
resume
We propose a one-dimensional Saint-Venant (also known as open channel or shallow water) equation model for overland flows including a water input--output source term. We derive the model from the two-dimensional Navier--Stokes equations under the shallow water assumption, with boundary conditions including recharge via ground infiltration and runoff. We show that the energy-consistency of the resulting Saint-Venant model is strictly dependent upon the assumed level of rain- or recharge-induced friction. The proposed model extends most extant models by adding more scope depending on friction terms that depend on the rate of water entering or exiting the flow via recharge and infiltration. The obtained entropy relation for our model validate it both mathematically and physically. We compare both models computationally based on a kinetic finite volume scheme; in particular, we provide numerical evidence that the two models may show drastically different results, where the model conditioned on the flow velocity provides what may not be the physically relevant solution because of the lack of the appropriate friction terms depending on a master parameter ($\alpha$). We also look at a comparison with real-life data.
DOI
DOI : 10.3390/mca26010001
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auteur
Mohamed Ali Debyaoui, Mehmet Ersoy
titre
Generalised Serre-Green-Naghdi equations for open channel and for natural river hydraulics
article
Asymptotic Analysis, In press, ⟨10.3233/ASY-201647⟩
resume
In this paper, we present a new non-linear dispersive model for open channel and river flows. These equations are the second order shallow water approximation of the section-averaged (three-dimensional) incompressible and irrotational Euler system. This new asymptotic model generalises the well-known one-dimensional Serre-Green-Naghdi (SGN) equations for rectangular section on uneven bottom to arbitrary chan-nel/river section.
DOI
DOI : 10.3233/ASY-201647
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auteur
Jean-Baptiste Clément, Damien Sous, Frederic Golay, Mehmet Ersoy
titre
Wave-driven Groundwater Flows in Sandy Beaches: A Richards Equation-based Model
article
Journal of Coastal Research, 2020, Proceedings from the 16th International Coastal Symposium, Special issue 95, pp.1047-1051. ⟨10.2112/SI95-204.1⟩
resume
A groundwater model is developed to simulate flows under the swash zone of sandy beach. Variably saturated beach is described by Richards equation, which is solved by the Rivage code thanks to a discontinuous Galerkin method. The SWASH code is used to simulate wave propagation in nearshore waters and compute suitable boundary conditions at the beachface to force Richards equation. An idealized case is considered to investigate the beach groundwater response to the action of swash cycles associated with long IG-like waves: infiltration/ exfiltration, water table, hydraulic head and pore velocities are outlined. A good qualitative agreement is found with observations from experimental studies in the literature.
DOI
DOI : 10.2112/SI95-204.1
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2016

auteur
Thomas Altazin, Mehmet Ersoy, Frederic Golay, Damien Sous, Lyudmyla Yushchenko
titre
Numerical investigation of BB-AMR scheme using entropy production as refinement criterion
article
International Journal of Computational Fluid Dynamics, 2016, ⟨10.1080/10618562.2016.1194977⟩
resume
In this work, a parallel finite volume scheme on unstructured meshes is applied to fluid flow for multidimensional hyperbolic system of conservation laws. It is based on a block-based adaptive mesh refinement strategy which allows quick meshing and easy parallelisation. As a continuation and as an extension of a previous work, the useful numerical density of entropy production is used as mesh refinement criterion combined with a local time-stepping method to preserve the computational time. Then, we numerically investigate its efficiency through several test cases with a confrontation with exact solution or experimental data.
DOI
DOI : 10.1080/10618562.2016.1194977
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auteur
Mehmet Ersoy
titre
Dimension reduction for compressible pipe flows including friction
article
Asymptotic Analysis, 2016, 98 (3), pp.237-255. ⟨10.3233/ASY-161367⟩
resume
We present the full derivation of a one dimensional " Saint-Venant " like equations for barotropic compressible pipe flows including friction. The one dimensional hyperbolic system is called γ−pressurized model where γ is the adiabatic constant. It is obtained through the three dimensional barotropic Navier-Stokes equations under " thin layer " assumptions as a first order approximation. Prescribing suitable boundary conditions, one can introduce a general friction law and then explicitly show its geometrical (w.r.t the hydraulic radius) and hydrodynamical (w.r.t the Oser number) dependencies in the reduced model. In particular, for linear pressure law (γ = 1), we justify the one dimensional pressurized model (called P-model) introduced by the author in the context of unsteady mixed flows in closed water pipes. For non linear pressure law (γ = 1), the γ−pressurized model describes the evolution of a compressible (almost) gravityless flow.
DOI
DOI : 10.3233/ASY-161367
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2015

auteur
Frederic Golay, Mehmet Ersoy, Lyudmyla Yushchenko, Damien Sous
titre
Block-based adaptive mesh refinement scheme using numerical density of entropy production for three-dimensional two-fluid flows
article
International Journal of Computational Fluid Dynamics, 2015, ⟨10.1080/10618562.2015.1012161⟩
resume
In this work, we present a fast and parallel finite volume scheme on unstructured meshes applied to complex fluid flow. The mathematical model is based on a three-dimensional compressible low Mach two-phase flows model, combined with a linearised ‘artificial pressure’ law. This hyperbolic system of conservation laws allows an explicit scheme, improved by a block-based adaptive mesh refinement scheme. Following a previous one-dimensional work, the useful numerical density of entropy production is used as mesh refinement criterion. Moreover, the computational time is preserved using a local time-stepping method. Finally, we show through several test cases the efficiency of the present scheme on two- and three-dimensional dam-break problems over an obstacle.
DOI
DOI : 10.1080/10618562.2015.1012161
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auteur
Lyudmyla Yushchenko, Frederic Golay, Mehmet Ersoy
titre
Entropy production and mesh refinement – Application to wave breaking
article
Mechanics & Industry, 2015, 16 (3), pp.5. ⟨10.1051/meca/2015003⟩
resume
We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. This approach is validated in the case of classical 1D numerical tests. Then, we present a 3D application. Indeed, according to an eulerian bi-fluid formulation at low Mach, an hyperbolic system of conservation laws allows an easily parallelization and mesh refinement by “blocks”.
DOI
DOI : 10.1051/meca/2015003
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2014

auteur
Christian Bourdarias, Mehmet Ersoy, Stéphane Gerbi
titre
Unsteady mixed flows in non uniform closed water pipes: a Full Kinetic Appraoch
article
Numerische Mathematik, 2014, 128 (2), pp.217-263. ⟨10.1007/s00211-014-0611-7⟩
resume
We recall the PFS model constructed for the modeling of unsteady mixed flows in closed water pipes where transition points between the free surface and pressurized flow are treated as a free boundary associated to a discontinuity of the gradient of pressure. Then we present a numerical kinetic scheme for the computations of unsteady mixed flows in closed water pipes. This kinetic method that we call FKA for ''Full Kinetic Approach" is an easy and mathematically elegant way to deal with multiple transition points when the changes of state between free surface and pressurized flow occur. We use two approaches namely the ''ghost waves approach'' and the ''Full Kinetic Approach" to treat these transition points. We show that this kinetic numerical scheme has the following properties: it is wet area conservative, under a CFL condition it preserves the wet area positive, it treats ''naturally'' the drying and flooding area and most of all it preserves every stationary flow. Finally numerical experiment versus laboratory experiment is presented and the scheme produces results that are in a very good agreement. We also present a numerical experiment when flooding and drying flows may occur and finally make a numerical study of the order of the kinetic method.
DOI
DOI : 10.1007/s00211-014-0611-7
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2013

auteur
Christian Bourdarias, Mehmet Ersoy, Stéphane Gerbi
titre
Air entrainment in transient flows in closed water pipes: a two-layer approach.
article
ESAIM: Mathematical Modelling and Numerical Analysis, 2013, 47 (2), pp.507-538. ⟨10.1051/m2an/2012036⟩
resume
In this paper, we first construct a model for free surface flows that takes into account the air entrainment by a system of four partial differential equations. We derive it by taking averaged values of gas and fluid velocities on the cross surface flow in the Euler equations (incompressible for the fluid and compressible for the gas). The obtained system is conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation of this system to finally construct a two-layer kinetic scheme in which a special treatment for the ''missing'' boundary condition is performed. Several numerical tests on closed water pipes are performed and the impact of the loss of hyperbolicity is discussed and illustrated. Finally, we make a numerical study of the order of the kinetic method in the case where the system is mainly non hyperbolic. This provides a useful stability result when the spatial mesh size goes to zero.
DOI
DOI : 10.1051/m2an/2012036
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auteur
Mehmet Ersoy, Frederic Golay, Lyudmyla Yushchenko
titre
Adaptive multi scale scheme based on numerical density of entropy production for conservation laws
article
Central European Journal of Mathematics, 2013, 11 (8), pp.1392-1415. ⟨10.2478/s11533-013-0252-6⟩
resume
We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solution remains smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. We also use high order time extensions using Adams-Bashforth time integration technique as well as a second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: the Sod's shock tube problem, the Lax's shock tube problem and the Shu-Osher test problem.
DOI
DOI : 10.2478/s11533-013-0252-6
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auteur
Mehmet Ersoy, Eduard Feireisl, Enrique Zuazua
titre
Sensitivity analysis of 1-d steady forced scalar conservation laws
article
Journal of Differential Equations, 2013, 254 (9), pp.3817-3834. ⟨10.1016/j.jde.2013.01.041⟩
resume
We analyze 1 − d forced steady state scalar conservation laws. We first show the existence and uniqueness of entropy solutions as limits as t → ∞ of the corresponding solutions of the scalar evolutionary hyperbolic conservation law. We then linearize the steady-state equation with respect to perturbations of the forcing term. This leads to a linear first order differential equation with, possibly, discontinuous coefficients. We show the existence and uniqueness of solutions in the context of duality solutions. We also show that this system corresponds to the steady state version of the linearized evolutionary hyperbolic conservation law. This analysis leads us to the study of the sensitivity of the shock location with respect to variations of the forcing term, an issue that is relevant in applications to optimal control and parameter identification problems.
DOI
DOI : 10.1016/j.jde.2013.01.041
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2012

auteur
Christian Bourdarias, Mehmet Ersoy, Stéphane Gerbi
titre
A mathematical model for unsteady mixed flows in closed water pipes
article
Science China Mathematics, 2012, 55 (2), pp.221-244. ⟨10.1007/s11425-011-4353-z⟩
resume
We present the formal derivation of a new unidirectional model for unsteady mixed flows in non uniform closed water pipe. In the case of free surface incompressible flows, the \FS-model is formally obtained, using formal asymptotic analysis, which is an extension to more classical shallow water models. In the same way, when the pipe is full, we propose the \Pres-model, which describes the evolution of a compressible inviscid flow, close to gas dynamics equations in a nozzle. In order to cope the transition between a free surface state and a pressured (i.e. compressible) state, we propose a mixed model, the \PFS-model, taking into account changes of section and slope variation.
DOI
DOI : 10.1007/s11425-011-4353-z
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auteur
Mehmet Ersoy, Timack Ngom
titre
Existence of a global weak solution to one model of Compressible Primitive Equations
article
Comptes Rendus. Mathématique, 2012, 350 (1), pp.379-382. ⟨10.1016/j.crma.2012.04.013⟩
resume
In this note, we show a global weak existence result for a two dimensionnal Compressible Primitive Equations for atmosphere dynamics modeling.
DOI
DOI : 10.1016/j.crma.2012.04.013
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2011

auteur
Christian Bourdarias, Stéphane Gerbi, Mehmet Ersoy
titre
A kinetic scheme for transient mixed flows in non uniform closed pipes: a global manner to upwind all the source terms
article
Journal of Scientific Computing, 2011, 48 (1-3), pp. 89-104. ⟨10.1007/s10915-010-9456-0⟩
resume
We present a numerical kinetic scheme for an unsteady mixed pressurised and free surface model. This model has a source term depending on both the space variable and the unknown, U, of the system. The source term is composed by a topography, a section variation, a curvature (also called corrective) and a friction term. Using the Finite Volume and Kinetic (FVK) framework, we propose an approximation of the source terms following the principle of interfacial upwind with a kinetic interpretation: the source term is not treated as a volumic term, but included in the numerical fluxes. Then, several numerical tests are presented.
DOI
DOI : 10.1007/s10915-010-9456-0
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auteur
Mehmet Ersoy, Timack Ngom, Mamadou Sy
titre
Compressible primitive equation: formal derivation and stability of weak solutions
article
Nonlinearity, 2011, 24 (1), ⟨10.1088/0951-7715/24/1/004⟩
resume
We present a formal derivation of a simplified version of Compressible Primitive Equations (CPEs) for atmosphere modeling. They are obtained from $3$-D compressible Navier-Stokes equations with an \emph{anisotropic viscous stress tensor} where viscosity depends on the density. We then study the stability of the weak solutions of this model by using an intermediate model, called model problem, which is more simple and practical, to achieve the main result.
DOI
DOI : 10.1088/0951-7715/24/1/004
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2009

auteur
Christian Bourdarias, Mehmet Ersoy, Stéphane Gerbi
titre
A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme
article
International Journal on Finite Volumes, 2009, 6 (2), pp.1-47
resume
We present the derivation of a new unidirectional model for We present the derivation of a new unidirectional model for unsteady mixed flows in non uniform closed water pipes. We introduce a local reference frame to take into account the local perturbation caused by the changes of section and slope. Then an asymptotic analysis is performed to obtain a model for free surface flows and another one for pressurized flows. By coupling these models through the transition points by the use of a common set of variables and a suitable pressure law, we obtain a simple formulation called PFS-model close to the shallow water equations with source terms. It takes into account the changes of section and the slope variation in a continuous way through transition points. Transition point between the two types of flows is treated as a free boundary associated to a discontinuity of the gradient of pressure. The numerical simulation is performed by making use of a Roe-like finite volume scheme that we adapted to take into account geometrical source terms in the convection matrix. Finally some numerical tests are presented.
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auteur
Christian Bourdarias, Mehmet Ersoy, Stéphane Gerbi
titre
A kinetic scheme for pressurized flows in non uniform pipes
article
Monografias de la Real Academia de Ciencias de Zaragoza, 2009, 31, pp.1-20
resume
The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes with variable sections. Firstly, we detail the derivation of the mathematical model in curvilinear coordinates under some hypothesis and we performe a formal asymptotic analysis. Then the obtained system is written as a conservative hyperbolic partial differential system of equations, and we recall how to obtain the corresponding kinetic formulation based on an upwinding of the source term due to the pseudo topography'' performed in a close manner described by Perthame and al.
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Conference papers

2020

auteur
Jean-Baptiste Clément, Frederic Golay, Mehmet Ersoy, Damien Sous
titre
Adaptive Discontinuous Galerkin Method for Richards Equation
article
Topical Problems of Fluid Mechanics 2020, Feb 2020, Prague, Czech Republic. pp.27-34, ⟨10.14311/TPFM.2020.004⟩
resume
This paper deals with the numerical simulation of time-dependant flows in partially saturated porous media modelled by Richards equation. A Discontinuous Galerkin method together with an implicit scheme are used to approximate the solution. The mathematical framework and a procedure for solving this nonlinear equation are presented. In particular, treatment of seepage boundary condition and adaptive time step are underlined. An adaptive mesh refinement technique is also outlined in order to capture wetting front efficiently. Some numerical simulations are performed to illustrate the performance of the method.
DOI
DOI : 10.14311/TPFM.2020.004
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2019

auteur
Jean-Baptiste Clément, Mehmet Ersoy, Frederic Golay, Damien Sous
titre
Discontinuous Galerkin Method for Steady-State Richards Equation
article
Topical Problems of Fluid Mechanics 2019, Feb 2019, Prague, Czech Republic. pp.53-62, ⟨10.14311/TPFM.2019.008⟩
resume
This work is devoted to the numerical simulation of flows in partially saturated porous media. We describe the Richards equation governing the subsurface flow and discuss its range of applicability. A discontinuous Galerkin formulation is used to approximate the steady-state Richards equation. To this end, we present the mathematical framework and a procedure for solving the nonlinear equation. Numerical tests are carried out to highlight properties of the discontinuous Galerkin method and a test case is compared to experimental data to validate the model.
DOI
DOI : 10.14311/TPFM.2019.008
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2015

auteur
Mehmet Ersoy
titre
Dimension reduction for incompressible pipe and open channel flow including friction
article
Conference Applications of Mathematics 2015, in honor of the 90th birthday of Ivo Babuška and 85th birthday of Milan Práger and Emil Vitásek , J. Brandts and S. Korotov and M. Křížek and K. Segeth and J. Ŝístek and T. Vejchodský, Nov 2015, Institute of Mathematics CAS, Prague, France. pp.17-33
resume
We present the full derivation of a one-dimensional free surface pipe or open channel flow model including friction with non constant geometry. The free surface model is obtained from the three-dimensional incompress-ible Navier-Stokes equations under shallow water assumptions with prescribed " well-suited " boundary conditions.
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2013

auteur
Lyudmyla Yushchenko, Frederic Golay, Mehmet Ersoy
titre
Entropy production and mesh refinement – application to wave breaking
article
21ème Congrès Français de Mécanique, Aug 2013, Bordeaux, France. ⟨10.1051/meca/2015003⟩
resume
We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. This approach is validated in the case of classical 1D numerical tests. Then, we present a 3D application. Indeed, according to an eulerian bi-fluid formulation at low Mach, an hyperbolic system of conservation laws allows an easily parallelization and mesh refinement by " blocks " .
DOI
DOI : 10.1051/meca/2015003
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auteur
Frederic Golay, Mehmet Ersoy, Lyudmyla Yushchenko
titre
Entropy production as a mesh refinement criterion: application to wave breaking
article
Topical Problems of Fluid Mechanics, Feb 2013, Prague, Czech Republic
resume
We propose an adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). Thus it is used as an a posteriori error which provides information on the need to refine the mesh in the regions where discontinuities occur and to coarsen the mesh in the regions where the solutions remain smooth. Nevertheless, due to the CFL stability condition the time step is restricted and leads to time consuming simulations. Therefore, we propose a local time stepping algorithm. We numerically investigate the efficiency of the scheme through several 1D test cases and 3D dam break test case.
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Book sections

2021

auteur
Mohamed Ali Debyaoui, Mehmet Ersoy
titre
A Generalised Serre-Green-Naghdi equations for variable rectangular open channel hydraulics and its finite volume approximation
article
Muñoz-Ruiz M.L., Parés C., Russo G. (eds) Recent Advances in Numerical Methods for Hyperbolic PDE Systems. SEMA SIMAI Springer Series, vol 28. Springer, 2021, 978-3-030-72850-2. ⟨10.1007/978-3-030-72850-2_11⟩
resume
We present a non-linear dispersive shallow water model which enters in the framework of section-averaged models. These new equations are derived up to the second order of the shallow water approximation starting from the three-dimensional incompressible and irrotational Euler system. The derivation is carried out in the case of non uniform rectangular section and it generalises the well-known one-dimensional Serre-Green-Naghdi (SGN) equations on uneven bottom. The obtained model is fully consistent with the Euler system. We propose a well-balanced finite volume approximation and we present some numerical results to show the influence of the section variation.
DOI
DOI : 10.1007/978-3-030-72850-2_11
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Habilitation à diriger des recherches

2020

auteur
Mehmet Ersoy
titre
From hydrostatic to non-hydrostatic models in fluid mechanics: modeling, mathematical and numerical analysis, and computational fluid dynamics
article
Numerical Analysis [math.NA]. Université De Toulon, 2020
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Poster communications

2022

auteur
Jean-Baptiste Clément, Frederic Golay, Damien Sous, Mehmet Ersoy
titre
Numerical simulation of flows in unsaturated porous media by an adaptive discontinuous Galerkin method: application to La Verne dam
article
Topical Problems of Fluid Mechanics 2022, Feb 2022, Prague, Czech Republic
resume
Richards' equation is used to simulate flows in unsaturated porous media but it remains challenging to solve cost-effectively for complex cases. An adaptive strategy is proposed to deal with sharp wetting fronts. It is based on weighted discontinuous Galerkin methods and adaptive mesh refinement. The reservoir impoundment of La Verne dam is a demanding case because there are several materials, stiff hydraulic properties, dynamic boundary condition and seepage. The comparison of results with experimental data highlights the potential of the solving strategy.
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Reports

2013

auteur
Mehmet Ersoy, Chiara Simeoni
titre
Consistency of the kinetic scheme with reflections
article
[Research Report] IMATH. 2013
resume
In this note, we study the consistency of the Finite Volume kinetic scheme with reflections which is well-known to preserve the main physical properties of hyperbolic systems with source term. In particular, we prove that the numerical fluxes are consistent with the exact flux and are asymptotically consistent with the source term.
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auteur
Mehmet Ersoy, Frederic Golay, Lyudmyla Yushchenko
titre
Adaptive scheme based on entropy production: robustness through severe test cases for hyperbolic conservation laws
article
[Research Report] Imath. 2013
resume
In this paper, we study the robustness of the 1D adaptive numerical scheme based on numerical density of entropy production for hyperbolic conservation. In particular, we focus on vacuum states or low densities states or both in presence of strong shocks solutions to the one-dimensional gas dynamics equations for ideal gas. To this purpose, it is numerically shown that the adaptive numerical scheme is well-designed to capture accurately these type of solutions as the pressure vanishes. The efficiency of the scheme is investigated through several test cases for first and second order scheme with a MUSCL reconstruction. We focus in particular on several numerical experiments to test the ability to conserve the entropy in the adaptive framework, to capture strong discontinuities (both shocks and contact dis- continuities) as well as strong rarefactions. We also perform some numerical test case for solutions which are close to the vacuum or vanishes.
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2009

auteur
Mehmet Ersoy
titre
A Construction of Biorthogonal Wavelets With a Compact Operator
article
[Research Report] Université de Savoie. 2009
resume
We present a construction of biorthogonal wavelets using a compact operator which allows to preserve or increase some properties: regularity/vanishing moments, parity, compact supported. We build then a simple algorithm which computes new filters.
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Theses

2010

auteur
Mehmet Ersoy
titre
Modélisation, analyse mathématique et numérique de divers écoulements compressibles ou incompressibles en couche mince.
article
Mathématiques [math]. Université de Savoie, 2010. Français. ⟨NNT : ⟩
resume
Dans la première partie, on dérive formellement les équations \PFS (\textbf{P}ressurised and \textbf{F}ree \textbf{S}urface) pour les écoulements mixtes en conduite fermée avec variation de géométrie. On écrit l'approximation de ces équations à l'aide d'un solveur VFRoe et d'un solveur cinétique en décentrant les termes sources aux interfaces. En particulier, on propose le décentrement d'un terme de friction, donnée par la loi de Manning-Strickler, en introduisant la notion de \emph{pente dynamique}. Enfin, on construit un schéma bien équilibré préservant les états stationnaires au repos en définissant une matrice à profil stationnaire conçue pour le schéma VFRoe. Suivant cette idée, on construit, en toute généralité, un schéma bien équilibré préservant tous les états stationnaires. Pour traiter les points de transitions (i.e. le changement de type d'écoulement surface libre vers charge et vice et versa), on étend la méthode des \og ondes fantômes\fg~ dans ce contexte et on propose un traitement complètement cinétique. Dans la deuxième partie, on étudie des équations primitives compressibles simplifiées dans le cadre de la modélisation de la dynamique de l'atmosphère. En particulier, on obtient un résultat d'existence de solutions faibles globales en temps en dimension $2$ d'espace. On établit également un résultat de stabilité de solutions faibles pour le modèle en dimension $3$ d'espace. À cet égard, on introduit un changement de variables convenable qui permet de transformer les équations initiales en un modèle plus simple à étudier. Dans la troisième et dernière partie, on présente une courte introduction à la cavitation. En particulier, on rappelle les différents types de cavitation et les modèles mathématiques de Rayleigh-Plesset pour l'étude d'une bulle isolée et un modèle de mélange plus complexe. En vue de modéliser la cavitation dans les conduites fermées, on introduit un modèle à deux couches pour prendre en compte, dans un premier temps, l'effet d'une poche d'air comprimée par la surface libre et les bords de la conduite. En particulier, le système obtenu, à $4$ équations, est généralement non hyperbolique et ses valeurs propres ne sont pas calculables explicitement. On propose alors une approximation numérique basée sur un schéma cinétique mono-couche. Dans le dernier chapitre, on dérive formellement un modèle de transport de sédiments basé sur l'équation de Vlasov couplée à des équations de Navier-Stokes compressibles avec un tenseur de viscosité anisotrope. Ce modèle est ensuite obtenu par le biais de deux analyses asymptotiques.
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Preprints, Working Papers, ...

2019

auteur
Kévin Pons, Mehmet Ersoy
titre
Adaptive mesh refinement method. Part 1: Automatic thresholding based on a distribution function
article
2019
resume
The accurate numerical simulation of large scale flows, together with the detailed modeling of flooding or drying of small-scale regions, is a difficult and a challenging problem. Adaptive mesh method allows, in principle, to solve accurately those scales. However in practice, on one hand, the lack of a priori or efficient a posteriori error estimates, especially for multidimensional hyperbolic problems, make the analysis harder. On the other hand, once a mesh refinement criterion is chosen, the difficult problem is to determine the mesh refinement threshold parameter which is certainly the most important part of the adaptive process. The smaller this parameter is, the higher the number of cells refined is at the expense of the computational cost. In this work, we numerically investigate different refinement criteria and we present a general procedure to determine automatically a mesh refinement threshold for any given mesh refinement criterion. To this end the decreasing rearrangement (distribution) function of the mesh refinement criterion is introduced to catch relevant scales. The efficiency of the automatic thresholding method is illustrated through the one dimensional Saint-Venant system. Multidimensional and real life applications such as Tsunamis propagations are dealt in the second part.
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2016

auteur
Kévin Pons, Richard Marcer, Mehmet Ersoy, Frederic Golay, Richard Marcer Principia
titre
Adaptive mesh refinement method. Part 2: Application to tsunamis propagation
article
2016
resume
Numerical simulations of multi dimensional large scale fluid-flows such as tsunamis, are still nowadays a challenging and a difficult problem. To this purpose, a parallel finite volume scheme on adaptive unstructured meshes for multi dimensional Saint-Venant system is presented. The adaptive mesh refinement method is based on a block-based decomposition (called BB-AMR) which allows quick meshing and easy parallelization. The main difficulty addressed here concerns the selection of the mesh refinement threshold which is certainly the most important parameter in the AMR method. Usually, the threshold is calibrated according to the test problem to balance the accuracy of the solution and the computational cost. To avoid " hand calibration " , we apply an automatic threshold method based on the decreasing rearrangement function of the mesh refinement criterion. This method is applied and validated successfully to the one and two dimensional non homogeneous Saint-Venant system through several tsunamis propagation test cases.
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