arXiv subjects

Christian Bourdarias

Publications and source records attributed to Christian Bourdarias.

13 recordsLinked to original sources

arXiv2024

Entropy solutions in $BV^s$ for a class of triangular systems involving a transport equation

In this article, we consider a class of strictly hyperbolic triangular systems involving a transport equation. Such systems are known to create measure solutions for the initial value problem. Adding a stronger transversality assumption on the fields, we are able to obtain solutions in $L^\infty$ under optimal fractional $BV$ regularity of the initial data. Our results show that the critical fractional regularity is $s=1/3$. We also construct an initial data that is not in $BV^{1/3}$ but for which a blow-up in $L^\infty$ occurs, proving the optimality of our results.

math.AP↗

arXiv2017

A numerical scheme for an improved Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves

In this paper we introduce a new reformulation of the Green-Naghdi model in the Camassa-Holm regime for the propagation of internal waves over a flat topography derived by Duchêne, Israwi and Talhouk. These new Green-Naghdi systems are adapted to improve the frequency dispersion of the original model, they share the same order of precision as the standard one but have an appropriate structure which makes them much more suitable for the numerical resolution. We develop a second order splitting scheme where the hyperbolic part of the system is treated with a high-order finite volume scheme and the dispersive part is treated with a finite difference approach. Numerical simulations are then performed to validate the model and the numerical methods.

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arXiv2013

Unsteady mixed flows in non uniform closed water pipes: a Full Kinetic Appraoch

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.

math.AP↗

arXiv2013

Fractional BV spaces and first applications to scalar conservation laws

The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some "fractional $BV$ spaces" denoted $BV^s$, for $0 < s \leq 1$, introduced by Love and Young in 1937. The $BV^s(\R)$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\R)$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with $BV^s$ initial data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes.

math.AP↗

arXiv2012

Air entrainment in transient flows in closed water pipes: a two-layer approach

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.

math.AP↗

arXiv2011

A mathematical model for unsteady mixed flows in closed water pipes

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.

math.AP↗

arXiv2010

A kinetic scheme for transient mixed flows in non uniform closed pipes: a global manner to upwind all the source terms

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.

math.AP↗

arXiv2010

Blow up at the hyperbolic boundary for a 2x2 system arising from chemical engineering

We consider an initial boundary value problem for a 2x2 system of conservation laws modeling heatless adsorption of a gaseous mixture with two species and instantaneous exchange kinetics, close to the system of Chromatography. In this model the velocity is not constant because the sorption effect is taken into account. Exchanging the roles of the x, t variables we obtain a strictly hyperbolic system with a zero eigenvalue. Our aim is to construct a solution with a velocity which blows up at the corresponding characteristic "hyperbolic boundary" {t = 0}.

math.AP↗

arXiv2009

A model for unsteady mixed flows in non uniform closed water pipes and a well-balanced finite volume scheme

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.

math.AP↗

arXiv2009

Strong Stability with respect to weak limit for a Hyperbolic System arising from Gas Chromatography

We investigate a system related to a particular isothermal gas-solid chromatography process, called ?Pressure Swing Adsorption?, with two species and instantaneous exchange kinetics. This system presents the particularity to have a linearly degenerate eigenvalue: this allows the velocity of the gaseous mixture to propagate high frequency waves. In case of smooth concentrations with a general isotherm, we prove L1 stability for concentrations with respect to weak limits of the inlet boundary velocity. Using the Front Tracking Algorithm (FTA), we prove a similar result for concentrations with bounded variation (BV) under some convex assumptions on the isotherms. In both cases we show that high frequency oscillations with large amplitude of the inlet velocity can propagate without affecting the concentrations.

math.AP↗

arXiv2009

A kinetic scheme for unsteady pressurised flows in closed water pipes

The aim of this paper is to present a kinetic numerical scheme for the computations of transient pressurised flows in closed water pipes. Firstly, we detail the mathematical model written as a conservative hyperbolic partial differentiel system of equations, and the we recall how to obtain the corresponding kinetic formulation. Then we build the kinetic scheme ensuring an upwinding of the source term due to the topography performed in a close manner described by Perthame et al. using an energetic balance at microscopic level for the Shallow Water equations. The validation is lastly performed in the case of a water hammer in a uniform pipe: we compare the numerical results provided by an industrial code used at EDF-CIH (France), which solves the Allievi equation (the commonly used equation for pressurised flows in pipes) by the method of characteristics, with those of the kinetic scheme. It appears that they are in a very good agreement.

math.AP↗

arXiv2008

A pseudo active kinematic constraint for a biological living soft tissue: an effect of the collagen network

Recent studies in mammalian hearts show that left ventricular wall thickening is an important mechanism for systolic ejection and that during contraction the cardiac muscle develops significant stresses in the muscular cross-fiber direction. We suggested that the collagen network surrounding the muscular fibers could account for these mechanical behaviors. To test this hypothesis we develop a model for large deformation response of active, incompressible, nonlinear elastic and transversely isotropic living soft tissue (such as cardiac or arteries tissues) in which we include a coupling effect between the connective tissue and the muscular fibers. Then, a three-dimensional finite element formulation including this internal pseudo-active kinematic constraint is derived. Analytical and finite element solutions are in a very good agreement. The numerical results show this wall thickening effect with an order of magnitude compatible with the experimental observations.

math.AP↗

arXiv2008

A kinetic scheme for pressurized flows in non uniform pipes

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.

math.AP↗