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42
Convergence of a nonmonotone scheme for HamiltonJacobiBellman equations with discontinuous data
, 2007
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An antidiffusive scheme for viability problems
 APPL. NUMER. MATH
, 2006
"... This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and the ..."
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Cited by 15 (6 self)
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This paper is concerned with the numerical approximation of viability kernels. The method described here provides an alternative approach to the usual viability algorithm. We first consider a characterization of the viability kernel as the value function of a related optimal control problem, and then use a specially relevant numerical scheme for its approximation. Since this value function is discontinuous, usual discretization schemes (such as finite differences) would provide a poor approximation quality because of numerical diffusion. Hence, we investigate the UltraBee scheme, particularly interesting here for its antidiffusive property in the transport of discontinuous functions. Although currently there is no available convergence proof for this scheme, we observed that numerically, the experiments done on several benchmark problems for computing viability kernels and capture basins are very encouraging compared to the viability algorithm, which fully illustrates the relevance of this scheme for numerical approximation of viability problems.
An efficient data structure and accurate scheme to solve front propagation problems
 J. Sci. Comput
"... In this paper, we are interested in some front propagation problems coming from control problems in ddimensional spaces, with d ≥ 2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an ..."
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Cited by 10 (8 self)
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In this paper, we are interested in some front propagation problems coming from control problems in ddimensional spaces, with d ≥ 2. As opposed to the usual level set method, we localize the front as a discontinuity of a characteristic function. The evolution of the front is computed by solving an HamiltonJacobiBellman equation with discontinuous data, discretized by means of the antidissipative Ultra Bee scheme. We develop an efficient dynamic storage technique suitable for handling front evolutions in large dimension. Then we propose a fast algorithm, showing its relevance on several challenging tests in dimension d = 2, 3, 4. We also compare our method with the techniques usually used in level set methods. Our approach leads to a computational cost as well as a memory allocation scaling as O(Nnb) in most situations, where Nnb is the number of grid nodes around the front. AMS Classification: 65M06, 49L99.
An explicit stable numerical scheme for the 1D transport equation, (Subm.), available at: http://hal.archivesouvertes.fr/hal00523197/fr
"... Abstract. We derive in this paper a numerical scheme in order to calculate solutions of 1D transport equations. This 2ndorder scheme is based on the method of characteristics and consists of two steps: The first step is about the approximation of the foot of the characteristic curve whereas the sec ..."
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Cited by 9 (9 self)
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Abstract. We derive in this paper a numerical scheme in order to calculate solutions of 1D transport equations. This 2ndorder scheme is based on the method of characteristics and consists of two steps: The first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two 2ndorder interpolation schemes so as to preserve the maximum principle. The resulting method is unconditionaly stable and designed for classical solutions but turns out to remain valid when shocks occur.
Simulation of fluidparticles flows: heavy particles, flowing regime and asymptoticpreserving schemes
 Commun. Math. Sci
"... Abstract. We are interested in an Eulerian–Lagrangian model describing particulate flows. The model under study consists of the Euler system and a VlasovFokkerPlanck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, ..."
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Cited by 8 (4 self)
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Abstract. We are interested in an Eulerian–Lagrangian model describing particulate flows. The model under study consists of the Euler system and a VlasovFokkerPlanck equation coupled through momentum and energy exchanges. This problem contains asymptotic regimes that make the coupling terms stiff, and lead to a limiting model of purely hydrodynamic type. We design a numerical scheme which is able to capture this asymptotic behavior without requiring prohibitive stability conditions. The construction of this Asymptotic Preserving scheme relies on an implicit discretization of the stiff terms which can be treated by efficient inversion methods. This method is a natural coupling of a kinetic solver for the particles with a kinetic scheme for the hydrodynamic Euler equations. Numerical experiments are conducted to study the performance of this scheme in various asymptotic regimes. Key words. Fluid–particles flows, hydrodynamic regimes, Asymptotic Preserving schemes, Kinetic schemes. AMS subject classifications. 82C80, 82C40, 35L65, 35Q35, 65M06, 76N15, 76M20.
Existence of global solutions to the 1D abstract bubble vibration model. Diffential Integral Equations
, 2013
"... The Abstract Bubble Vibration model (Abv) is a set of two PDEs consisting of a transport equation and a Poisson equation. It has been derived in order to provide a better understanding of hyperbolicelliptic couplings which are involved in a more general low Mach number system modelling bubbles. Wh ..."
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Cited by 6 (5 self)
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The Abstract Bubble Vibration model (Abv) is a set of two PDEs consisting of a transport equation and a Poisson equation. It has been derived in order to provide a better understanding of hyperbolicelliptic couplings which are involved in a more general low Mach number system modelling bubbles. While a local existence theorem has already been proven in any dimension for theAbvmodel, we get interested in this paper in the onedimensional case, where we prove the existence of global solutions no matter how smooth the data. We also provide explicit expressions of the solutions thanks to the method of characteristics that we apply to the transport equation despite the coupling with the Poisson equation. We then compute these solutions by means of a numerical scheme based on a discrete version of the method of characteristics.
A convergent and conservative schemes for nonclassical solutions based on kinetic relations
 I. Interfaces and Free Bound
, 2008
"... AbstractWe propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main ..."
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Cited by 5 (2 self)
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AbstractWe propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces, contrary to standard finite difference schemes. The main challenge is to achieve, at the discretization level, a consistency property with respect to a prescribed kinetic relation. The latter is required for the selection of physically meaningful nonclassical shocks. Our method is based on a reconstruction technique performed in each computational cell that may contain a nonclassical shock. To validate this approach, we establish several consistency and stability properties, and we perform careful numerical experiments. The convergence of the algorithm toward the physically meaningful solutions selected by a kinetic relation is demonstrated numerically for several test cases, including concaveconvex as well as convexconcave fluxfunctions. Résume ́ Nous proposons un nouvel algorithme pour approcher les solutions non classiques de lois de conservation hyperboliques. Le schéma aux différences finies présente ́ ici est conservatif et transporte de manière exacte les chocs non classiques, a ̀ la différences des algorithmes standard. La principale difficulte ́ est de garantir, au niveau discret, la consistance avec une re
Splitting schemes for the simulation of non equilibrium radiative flows
, 2006
"... Abstract. This paper is devoted to the numerical investigation of radiative hydrodynamics equations. We focus on non–equilibrium regimes and we design asymptotic preserving schemes which can handle the corresponding stiff equations. Our study includes relativistic effects and Doppler corrections. ..."
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Cited by 5 (2 self)
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Abstract. This paper is devoted to the numerical investigation of radiative hydrodynamics equations. We focus on non–equilibrium regimes and we design asymptotic preserving schemes which can handle the corresponding stiff equations. Our study includes relativistic effects and Doppler corrections.
The Vofire method for multicomponent flows on unstructured meshes
"... Résumé Nous proposons une nouvelle méthode numérique pour le calcul approché de solutions d’équations de transport, en dimension quelconque et sur maillages quelconques. Cet algorithme, de Volumes Finis, repose sur une méthode de reconstruction de l’inconnue dans chaque maille, où l’inconnue reconst ..."
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Cited by 2 (0 self)
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Résumé Nous proposons une nouvelle méthode numérique pour le calcul approché de solutions d’équations de transport, en dimension quelconque et sur maillages quelconques. Cet algorithme, de Volumes Finis, repose sur une méthode de reconstruction de l’inconnue dans chaque maille, où l’inconnue reconstruite ne varie que dans les directions orthogonales à la vitesse de transport. Ceci permet de générer un schéma non diffusif et stable. Utilisé pour le transport de fonctions indicatrices, il est ensuite incorporé à un algorithme Lagrangien avec remaillage pour le calcul d’écoulements multimatériaux compressibles. Abstract A new numerical method for the approximation of transport equations in any dimension on general meshes is here described. The algorithm relies on a reconstruction of the unknown in each cell, where the recontructed function varies only in the directions orthogonal to the velocity. This allows to elaborate a nondissipative and stable scheme for the transport of characteristic functions. It is then incorporated in a Lagrangian scheme with remeshing for the computation of multicomponent compressible flows. 1
Simulations of the LifshitzSlyozov equations: the role of coagulation terms in the asymptotic behavior
, 2012
"... We consider the LifshitzSlyozov system that describes the kinetics of precipitation from supersaturated solid solutions. We design specific Finite Volume schemes and we investigate numerically the behavior of the solutions, in particular the large time asymptotics. Our purpose is twofold: first, w ..."
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Cited by 1 (0 self)
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We consider the LifshitzSlyozov system that describes the kinetics of precipitation from supersaturated solid solutions. We design specific Finite Volume schemes and we investigate numerically the behavior of the solutions, in particular the large time asymptotics. Our purpose is twofold: first, we introduce an adapted scheme based on downwinding techniques in order to reduce the numerical diffusion; second, we discuss the influence of coagulation effects on the selection of the asymptotic profile. MSC Classification Number: 65M08 65R20 82C05 35L60 45K05 82D60 82C26 1