Publications

2014

• Quick reachability and proper extension for problems with unbounded controls
  
  • Aronna Maria Soledad
  • Motta Monica
  • Rampazzo Franco
  , 2014. For a CONTROL SYSTEM of the form _ x = f (x; u; v) + Σm =1 g (x)u_ ; on [0;T]; (x; u)(0) = ( x; u); with x : [0;T] ! IRn; u : [0;T] ! U IRm; v : [0;T] ! V IRl ; we rely on the notion of LIMIT SOLUTION, and we investigate whether minimum problems with L1controls are PROPER EXTENSIONS of regular problems with more regular controls (AC or BV). Motivation: optimality conditions, numerical methods, etc.
• Beyond first-order finite element schemes in micromagnetics
  
  • Kritsikis E.
  • Vaysset A.
  • Buda-Prejbeanu L.D.
  • Alouges F.
  • Toussaint Jean-Christophe
  Journal of Computational Physics, Elsevier, 2014, 256, pp.357. (10.1016/j.jcp.2013.08.035)
  DOI : 10.1016/j.jcp.2013.08.035
• On certain hyperelliptic signals that are natural controls for nonholonomic motion planning
  
  • Gauthier Jean-Paul
  • Monroy-Perez Felipe
  , 2014. In this paper we address the general problem of approximating, in a certain optimal way, non admissible motions of a kinematic system with nonholonomic constraints. Since this kind of problems falls into the general subriemannian geometric setting, it is natural to consider optimality in the sense of approximating by means of subriemannian geodesics. We consider sys-tems modeled by a subriemannian Goursat structure, a particular case being the well known system of a car with trailers, along with the associated parallel parking problem. Several authors approximate the successive Lie brackets by using trigonometric functions. By contrast, we show that the more natural op-timal motions are related with closed hyperelliptic plane curves with a certain number of loops.
• A generalized formulation of the Linear Sampling Method with exact characterization of targets in terms of farfield measurements
  
  • Audibert Lorenzo
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2014, 30 (035011). We propose and analyze a new formulation of the Linear Sampling Method that uses an exact characterization of the targets shape in terms of the so-called farfield operator (at a fixed frequency). This characterization is based on constructing nearby solutions of the farfield equation using minimizing sequences of a least squares cost functional with an appropriate penalty term. We first provide a general framework for the theoretical foundation of the method in the case of noise-free and noisy measurements operator. We then explicit applications for the case of inhomogeneous inclusions and indicate possible straightforward generalizations. We finally validate the method through some numerical tests and compare the performances with classical LSM and the factorization methods. (10.1088/0266-5611/30/3/035011)
  DOI : 10.1088/0266-5611/30/3/035011
• Weighted Radon transforms and first order differential systems on the plane
  
  • Novikov Roman
  Moscow Mathematical Journal, Independent University of Moscow, 2014, 14 (4), pp.807–823. We consider weighted Radon transforms on the plane, where weights are given as finite Fourier series in angle variable. By means of additive Riemann-Hilbert problem techniques, we reduce inversion of these transforms to solving first order differential systems on $\R^2=\C$ with a decay condition at infinity. As a corollary, we obtain new injectivity and inversion results for weighted Radon transforms on the plane.
• Two-dimensional von Neumann--Wigner potentials with a multiple positive eigenvalue
  
  • Novikov Roman
  • Taimanov Iskander
  • Tsarev Sergey
  Functional Analysis and Its Applications, Springer Verlag, 2014, 48 (4), pp.295-297. By the Moutard transformation method we construct two-dimensional Schrodinger operators with real smooth potential decaying at infinity and with a multiple positive eigenvalue. These potentials are rational functions of spatial variables and their sines and cosines.
• Transmission conditions on interfaces for Hamilton-Jacobi-Bellman equations
  
  • Rao Zhiping
  • Siconolfi Antonio
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2014, 257 (11), pp.3978--4014. We establish a comparison principle for a Hamilton-Jacobi-Bellman equation, more appropriately a system, related to an infinite horizon problem in presence of an interface. Namely a low dimensional subset of the state variable space where discontinuities in controlled dynamics and costs take place. Since corresponding Hamiltonians, at least for the subsolution part, do not enjoy any semicontinuity property, the comparison argument is rather based on a separation principle of the controlled dynamics across the interface. For this, we essentially use the notion of "-partition and minimal "-partition for intervals of definition of an integral trajectory. (10.1016/j.jde.2014.07.015)
  DOI : 10.1016/j.jde.2014.07.015
• Geometric Control Theory and sub-Riemannian Geometry
  
  • Stefani Gianna
  • Boscain Ugo
  • Gauthier Jean-Paul
  • Sarychev Andrey
  • Sigalotti Mario
  , 2014, pp.372. This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.
• The $\Gamma$-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension
  
  • Bellettini Giovanni
  • Chambolle Antonin
  • Goldman Michael
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014. In this paper we generalize to arbitrary dimensions a one-dimensional equicoerciveness and $\Gamma$-convergence result for a second derivative perturbation of Perona-Malik type functionals. Our proof relies on a new density result in the space of special functions of bounded variation with vanishing diffuse gradient part. This provides a direction of investigation to derive approximation for functionals with discontinuities penalized with a ''cohesive'' energy, that is, whose cost depends on the actual opening of the discontinuity.
• Role of non-ideality for the ion transport in porous media: derivation of the macroscopic equations using upscaling
  
  • Allaire Grégoire
  • Brizzi Robert
  • Dufrêche Jean-François
  • Mikelic Andro
  • Piatnitski Andrey
  Physica D: Nonlinear Phenomena, Elsevier, 2014, 282, pp.39-60. This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account finite size ions and screening effects. We first consider equilibrium solutions in the absence of external forces. In such a case, the velocity and diffusive fluxes vanish and the equilibrium electrostatic potential is the solution of a variant of Poisson-Boltzmann equation coupled with algebraic equations. Contrary to the ideal case, this nonlinear equation has no monotone structure. However, based on invariant region estimates for Poisson-Boltzmann equation and for small characteristic value of the solute packing fraction, we prove existence of at least one solution. To our knowledge this existence result is new at this level of generality. When the motion is governed by a small static electric field and a small hydrodynamic force, we generalize O'Brien's argument to deduce a linearized model. Our second main result is the rigorous homogenization of these linearized equations and the proof that the effective tensor satisfies Onsager properties, namely is symmetric positive definite. We eventually make numerical comparisons with the ideal case. Our numerical results show that the MSA model confirms qualitatively the conclusions obtained using the ideal model but there are quantitative differences arising that can be important at high charge or high concentrations. (10.1016/j.physd.2014.05.007)
  DOI : 10.1016/j.physd.2014.05.007
• Higher level molecular phylogeny of darkling beetles (Coleoptera: Tenebrionidae)
  
  • Kergoat Gael G.
  • Soldati Laurent L.
  • Clamens Anne Laure
  • Jourdan Hervé
  • Jabbour-Zahab Roula
  • Genson Guénaëlle
  • Bouchard Patrice
  • Condamine Fabien
  Systematic Entomology, Wiley-Blackwell, 2014, 39 (3), pp.486-499. Insect diversity represents about 60% of the estimated million-and-a-half described eukaryotic species worldwide, yet comprehensive and well-resolved intra-ordinal phylogenies are still lacking for the majority of insect groups. This is the case especially for the most species-rich insect group, the beetles (Coleoptera), a group for which less than 4% of the known species have had their DNA sequenced. In this study, we reconstruct the first higher level phylogeny based on DNA sequence data for the species-rich darkling beetles, a family comprising at least 20000 species. Although amongst all families of beetles Tenebrionidae ranks seventh in terms of species diversity, the lack of knowledge on the phylogeny and systematics of the group is such that its monophyly has been questioned (not to mention those of the subfamilies and tribes contained within it). We investigate the evolutionary history of Tenebrionidae using multiple phylogenetic inference methods (Bayesian inference, maximum likelihood and parsimony) to analyse a dataset consisting of eight gene fragments across 404 taxa (including 250 tenebrionid species). Although the resulting phylogenetic framework only encompasses a fraction of the known tenebrionid diversity, it provides important information on their systematics and evolution. Whatever the methods used, our results provide strong support for the monophyly of the family, and highlight the likely paraphyletic or polyphyletic nature of several important tenebrionid subfamilies and tribes, notably the polyphyletic subfamilies Diaperinae and Tenebrioninae that clearly require substantial revision in the future. Some interesting associations in several groups are also revealed by the phylogenetic analyses, such as the pairing of Aphtora Bates with Phrenapatinae. Furthermore this study advances our knowledge of the evolution of the group, providing novel insights into much-debated theories, such as the apparent relict distribution of the tribe Elenophorini. (10.1111/syen.12065)
  DOI : 10.1111/syen.12065
• Unsupervised Segmentation of Spectral Images with a Spatialized Gaussian Mixture Model and Model Selection
  
  • Cohen Serge X.
  • Le Pennec E.
  Oil & Gas Science and Technology - Revue d'IFP Energies nouvelles, Institut Français du Pétrole (IFP), 2014, 69 (2), pp.245-259. In this article, we describe a novel unsupervised spectral image segmentation algorithm. This algorithm extends the classical Gaussian Mixture Model-based unsupervised classification technique by incorporating a spatial flavor into the model: the spectra are modelized by a mixture of K classes, each with a Gaussian distribution, whose mixing proportions depend on the position. Using a piecewise constant structure for those mixing proportions, we are able to construct a penalized maximum likelihood procedure that estimates the optimal partition as well as all the other parameters, including the number of classes. We provide a theoretical guarantee for this estimation, even when the generating model is not within the tested set, and describe an efficient implementation. Finally, we conduct some numerical experiments of unsupervised segmentation from a real dataset. (10.2516/ogst/2014013)
  DOI : 10.2516/ogst/2014013
• Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist
  
  • Boscain U.
  • Chertovskih R. A.
  • Gauthier Jean-Paul
  • Remizov A. O.
  SIAM Journal on Imaging Sciences, Society for Industrial and Applied Mathematics, 2014, 7 (2), pp.669–695. (10.1137/130924731)
  DOI : 10.1137/130924731
• Stochastic Approximation Finite Element method: analytical formulas for multidimensional diffusion process
  
  • Bompis Romain
  • Gobet Emmanuel
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2014, 52 (6), pp.3140-3164. We derive an analytical weak approximation of a multidimensional diffusion process as coefficients or time are small. Our methodology combines the use of Gaussian proxys to approximate the law of the diffusion and a Finite Element interpolation of the terminal function applied to the diffusion. We call this method Stochastic Approximation Finite Element (SAFE for short) method. We provide error bounds of our global approximation depending on the diffusion process coefficients, the time horizon and the regularity of the terminal function. Then we give estimates of the computational cost of our algorithm. This shows an improved efficiency compared to Monte-Carlo methods in small and medium dimensions (up to 10), which is confirmed by numerical experiments. (10.1137/130928431)
  DOI : 10.1137/130928431
• The Factorization Method for a Cavity in an Inhomogeneous Medium
  
  • Meng Shixu
  • Haddar Houssem
  • Cakoni Fioralba
  Inverse Problems, IOP Publishing, 2014, 30 (045008). We consider the inverse scattering problem for a cavity that is bounded by a penetrable anisotropic inhomogeneous medium of compact support and seek to determine the shape of the cavity from internal measurements on a curve or surface inside the cavity. We derive a factorization method which provides a rigorous characterization of the support of the cavity in terms of the range of an operator which is computable from the measured data. The support of the cavity is determined without a-priori knowledge of the constitutive parameters of the surrounding anisotropic medium provided they satisfy appropriate physical as well as mathematical assumptions imposed by our analysis. Numerical examples are given showing the viability of our method. (10.1088/0266-5611/30/4/045008)
  DOI : 10.1088/0266-5611/30/4/045008
• A finite elements method to solve the Bloch–Torrey equation applied to diffusion magnetic resonance imaging
  
  • Nguyen Dang Van
  • Li Jing-Rebecca
  • Grebenkov Denis S
  • Le Bihan Denis
  Journal of Computational Physics, Elsevier, 2014, pp.283–302. The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses in a heterogeneous medium can be modeled by the multiple compartment Bloch-Torrey partial differential equation (PDE). In addition, steady-state Laplace PDEs can be formulated to produce the homogenized diffusion tensor that describes the diffusion characteristics of the medium in the long time limit. In spatial domains that model biological tissues at the cellular level, these two types of PDEs have to be completed with permeability conditions on the cellular interfaces. To solve these PDEs, we implemented a finite elements method that allows jumps in the solution at the cell interfaces by using double nodes. Using a transformation of the Bloch-Torrey PDE we reduced oscillations in the searched-for solution and simplified the implementation of the boundary conditions. The spatial discretization was then coupled to the adaptive explict Runge-Kutta-Chebychev time-stepping method. Our proposed method is second order accurate in space and second order accurate in time. We implemented this method on the FEniCS C++ platform and show time and spatial convergence results. Finally, this method is applied to study some relevant questions in diffusion MRI. (10.1016/j.jcp.2014.01.009)
  DOI : 10.1016/j.jcp.2014.01.009
• Asymptotic analysis of the transmission eigenvalue problem for a Dirichlet obstacle coated by a thin layer of non-absorbing media
  
  • Cakoni Fioralba
  • Haddar Houssem
  • Chaulet Nicolas
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2014, pp.36. We consider the transmission eigenvalue problem for an impenetrable obstacle with Dirichlet boundary condition surrounded by a thin layer of non-absorbing inhomogeneous material. We derive a rigorous asymptotic expansion for the first transmission eigenvalue with respect to the thickness of the thin layer. Our convergence analysis is based on a Max–Min principle and an iterative approach which involves estimates on the corresponding eigenfunctions. We provide explicit expressions for the terms in the asymptotic expansion up to order 3. (10.1093/imamat/hxu045)
  DOI : 10.1093/imamat/hxu045
• Integrative taxonomy of New Caledonian beetles: species delimitation and definition of the [i]Uloma isoceroides[/i] species group (Coleoptera, Tenebrionidae, Ulomini), with the description of four new species
  
  • Soldati Laurent
  • Kergoat Gael
  • Clamens Anne Laure
  • Jourdan Hervé
  • Jabbour-Zahab Roula
  • Condamine Fabien L.
  Zookeys, Pensoft, 2014, 415, pp.133-167. New Caledonia is an important biodiversity hotspot with much undocumented biodiversity, especially in many insect groups. Here we used an integrative approach to explore species diversity in the tenebrionid genus Uloma (Coleoptera, Tenebrionidae, Ulomini), which encompasses about 150 species, of which 22 are known from New Caledonia. To do so, we focused on a morphologically homogeneous group by comparing museum specimens with material collected during several recent field trips. We also conducted molecular phylogenetic analyses based on a concatenated matrix of four mitochondrial and three nuclear genes for 46 specimens. The morphological study allowed us to discover and describe four new species that belong to the group of interest, the Uloma isoceroides group. Molecular analyses confirmed the species boundaries of several of the previously described species and established the validity of the four new species. The phylogenetic analyses also provided additional information on the evolutionary history of the group, highlighting that a species that was thought to be unrelated to the group was in fact a member of the same evolutionary lineage. Molecular species delimitation confirmed the status of the sampled species of the group and also suggested some hidden (cryptic) biodiversity for at least two species of the group. Altogether this integrative taxonomic approach has allowed us to better define the boundaries of the Uloma isoceroides species group, which comprises at least 10 species: Uloma isoceroides (Fauvel, 1904), Uloma opacipennis (Fauvel, 1904), Uloma caledonica Kaszab, 1982, Uloma paniei Kaszab, 1982, Uloma monteithi Kaszab, 1986, Uloma robusta Kaszab, 1986, Uloma clamensae sp. n., Uloma condaminei sp. n., Uloma jourdani sp. n., and Uloma kergoati sp. n. We advocate more studies on other New Caledonian groups, as we expect that much undocumented biodiversity can be unveiled through the use of similar approaches (10.3897/zookeys.415.6623)
  DOI : 10.3897/zookeys.415.6623
• Optimal control of leukemic cell population dynamics
  
  • Dupuis Xavier
  Mathematical Modelling of Natural Phenomena, EDP Sciences, 2014, 9 (1), pp.4-26. We are interested in optimizing the co-administration of two drugs for some acute myeloid leukemias (AML), and we are looking for in vitro protocols as a first step. This issue can be formulated as an optimal control problem. The dynamics of leukemic cell populations in culture is given by age-structured partial differential equations, which can be reduced to a system of delay differential equations, and where the controls represent the action of the drugs. The objective function relies on eigenelements of the uncontrolled model and on general relative entropy, with the idea to maximize the efficiency of the protocols. The constraints take into account the toxicity of the drugs. We present in this paper the modeling aspects, as well as theoretical and numerical results on the optimal control problem that we get. (10.1051/mmnp/20149102)
  DOI : 10.1051/mmnp/20149102
• Local properties of almost-Riemannian structures in dimension 3
  
  • Boscain Ugo
  • Charlot Grégoire
  • Gaye Moussa
  • Mason Paolo
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2014, 35 (9). A 3D almost-Riemannian manifold is a generalized Riemannian manifold defined locally by 3 vector fields that play the role of an orthonormal frame, but could become collinear on some set $\Zz$ called the singular set. Under the Hormander condition, a 3D almost-Riemannian structure still has a metric space structure, whose topology is compatible with the original topology of the manifold. Almost-Riemannian manifolds were deeply studied in dimension 2. In this paper we start the study of the 3D case which appear to be reacher with respect to the 2D case, due to the presence of abnormal extremals which define a field of directions on the singular set. We study the type of singularities of the metric that could appear generically, we construct local normal forms and we study abnormal extremals. We then study the nilpotent approximation and the structure of the corresponding small spheres. We finally give some preliminary results about heat diffusion on such manifolds.
• Faster Speciation and Reduced Extinction in the Tropics Contribute to the Mammalian Latitudinal Diversity Gradient
  
  • Rolland Jonathan
  • Condamine Fabien L.
  • Jiguet Frederic
  • Morlon Hélène
  PLoS Biology, Public Library of Science, 2014, 12 (1), pp.e1001775. The increase in species richness from the poles to the tropics, referred to as the latitudinal diversity gradient, is one of the most ubiquitous biodiversity patterns in the natural world. Although understanding how rates of speciation and extinction vary with latitude is central to explaining this pattern, such analyses have been impeded by the difficulty of estimating diversification rates associated with specific geographic locations. Here, we use a powerful phylogenetic approach and a nearly complete phylogeny of mammals to estimate speciation, extinction, and dispersal rates associated with the tropical and temperate biomes. Overall, speciation rates are higher, and extinction rates lower, in the tropics than in temperate regions. The diversity of the eight most species-rich mammalian orders (covering 92% of all mammals) peaks in the tropics, except that of the Lagomorpha (hares, rabbits, and pikas) reaching a maxima in northern-temperate regions. Latitudinal patterns in diversification rates are strikingly consistent with these diversity patterns, with peaks in species richness associated with low extinction rates (Primates and Lagomorpha), high speciation rates (Diprotodontia, Artiodactyla, and Soricomorpha), or both (Chiroptera and Rodentia). Rates of range expansion were typically higher from the tropics to the temperate regions than in the other direction, supporting the ''out of the tropics'' hypothesis whereby species originate in the tropics and disperse into higher latitudes. Overall, these results suggest that differences in diversification rates have played a major role in shaping the modern latitudinal diversity gradient in mammals, and illustrate the usefulness of recently developed phylogenetic approaches for understanding this famous yet mysterious pattern. (10.1371/journal.pbio.1001775)
  DOI : 10.1371/journal.pbio.1001775
• VWAP execution and guaranteed VWAP
  
  • Guéant Olivier
  • Guillaume Royer
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2014, 5 (1), pp.445-471. If optimal liquidation using VWAP strategies has been considered in the literature, it has never been considered in the presence of permanent market impact and only rarely with execution costs. Moreover, only VWAP strategies have been studied and no pricing of guaranteed VWAP contract is provided. In this article, we develop a model to price guaranteed VWAP contracts in the most general framework for market impact. Numerical applications are also provided. (10.1137/130924676)
  DOI : 10.1137/130924676
• Numerical study of a macroscopic finite pulse model of the diffusion MRI signal
  
  • Li Jing-Rebecca
  • Nguyen Hang Tuan
  • Nguyen Dang Van
  • Haddar Houssem
  • Coatléven Julien
  • Le Bihan Denis
  Journal of Magnetic Resonance, Elsevier, 2014, pp.54–65. Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. The dMRI signal from a heterogeneous sample includes the contribution of the water proton magnetization from all spatial positions in a voxel. If the voxel can be spatially divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be approximated using the macroscopic Karger model, which is a system of coupled ordinary differential equations (ODEs), under the assumption that the duration of the diffusion-encoding gradient pulses is short compared to the diffusion time (the narrow pulse assumption). \soutnew{Recently, a new macroscopic ODE model of the dMRI signal, the Finite Pulse ODE (FP-ODE) model, was derived from the Bloch-Torrey partial differential equation (PDE), without the narrow pulse restriction, using periodic homogenization techniques.}{Recently, a new macroscopic model of the dMRI signal, without the narrow pulse restriction, was derived from the Bloch-Torrey partial differential equation (PDE) using periodic homogenization techniques.} \soutnew{When restricted to narrow pulses, the FP-ODE model has the same form as the Karger model.}{When restricted to narrow pulses, this new homogenized model has the same form as the Karger model.} We conduct a numerical study of the \soutnew{FP-ODE}{new homogenized} model for voxels that are made up of periodic copies of a representative volume that contains spherical and cylindrical cells of various sizes and orientations and show that the signal predicted by the \soutnew{FP-ODE}{new} model approaches the reference signal obtained by solving the full Bloch-Torrey PDE in $O(\veps^2)$, where $\veps$ is the ratio between the size of the representative volume and \soutnew{the diffusion displacement}{a measure of the diffusion length}. When the narrow gradient pulse assumption is not satisfied, the \soutnew{FP-ODE}{new homogenized} model offers a much better approximation of the full PDE signal than the Karger model. Finally, preliminary results of applying the \soutnew{FP-ODE}{new} model to a voxel that is not made up of periodic copies of a representative volume are shown and discussed. (10.1016/j.jmr.2014.09.004)
  DOI : 10.1016/j.jmr.2014.09.004
• Complexity in control-affine systems
  
  • Jean Frédéric
  • Prandi Dario
  , 2014. We will consider affine-control systems, i.e., systems in the form _ q(t) = f0(q(t)) + Xm i=1 ui (t)fi (q(t)) Here, the point q belongs to a smooth manifold M the fi 's are smooth vector fields on M u 2 L1([0;T];Rm) This type of system appears in many applications Mechanical systems Quantum control Microswimmers (Tucsnak, Alouges) Neuro-geometry of vision (Mumfor, Petitot)
• A Robust Entropy-Satisfying Finite Volume Scheme for the Isentropic Baer-Nunziato Model
  
  • Coquel Frédéric
  • Hérard Jean-Marc
  • Saleh Khaled
  • Seguin Nicolas
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2014, 48 (1), pp.165-206. 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. (10.1051/m2an/2013101)
  DOI : 10.1051/m2an/2013101