Publications

Thèse de Doctorat de l'Université Pierre et Marie Curie - Paris 6.
Mathematical and numerical study of nonlinear hyperbolic equations: model coupling and nonclassical shocks.
Under the supervision of P.G. LeFloch, at the CEA de Saclay and Laboratoire J.-L. Lions.
Defended on Nov. 27th 2009.
[Abstract]
This thesis concerns the mathematical and numerical study of nonlinear hyperbolic partial differential equations. A first part deals with an emergent problematic: the coupling of hyperbolic equations. The pursued applications are linked with the mathematical coupling of computing platforms, dedicated to an adaptative simulation of multi-scale phenomena. We propose and analyze a new coupling formalism based on extended PDE systems avoiding the geometric treatment of the interfaces. In addition, it allows to formulate the problem in a multidimensional setting, with possible covering of the coupled models. This formalism allows in particular to equip the coupling procedure with viscous regularization mechanisms, useful in the selection of natural discontinuous solutions. We analyze existence and uniqueness in the framework of a parabolic regularization à la Dafermos. Existence of a solution holds true under very general conditions but failure of uniqueness may naturally arise as soon as resonance occurs at the interfaces. Next, we highlight that our extended PDE framework gives rise to another regularization strategy based on thick interfaces. In this setting, we prove existence and uniqueness of the solutions of the Cauchy problem for initial data in \(L^\infty\). The main tool consists in the derivation of a flexible and robust finite volume method for general triangulation which is analyzed in the setting of entropy measure-valued solutions by DiPerna. The second part is devoted to the definition of a finite volume scheme for the computing of nonclassical solutions of a scalar conservation law based on a kinetic relation. This scheme offers the feature to be stricto sensu conservative, in opposition to a Glimm approach that is only statistically conservative. The validity of our approach is illustrated through numerical examples.

Boutin B., Coquel F., and LeFloch P.G.,
Coupling techniques for nonlinear hyperbolic equations. III. A regularization method based on thick interfaces.
SIAM J. Numer. Anal., Vol. 51, 1108-1133 (2013).

[Abstract]
We continue our analysis of the coupling between nonlinear hyperbolic problems across possibly resonant interfaces. In the first two parts of this series, we introduced a new framework for coupling problems which is based on the so–called thin interface model and uses an augmented formulation and an additional unknown for the interface location; this framework has the advantage of avoiding any explicit modeling of the interface structure. In the present paper, we pursue our investigation of the augmented formulation and we introduce a new coupling framework which is now based on the so–called thick interface model. For scalar nonlinear hyperbolic equations in one space variable, we observe that the Cauchy problem is well-posed. Then, our main achievement in the present paper is the design of a new well-balanced finite volume scheme which is adapted to the thick interface model, together with a proof of its convergence toward the unique entropy solution (for a broad class of nonlinear hyperbolic equations). Due to the presence of a possibly resonant interface, the standard technique based on a total variation estimate does not apply, and DiPerna’s uniqueness theorem must be used. Following a method proposed by Coquel and LeFloch, our proof relies on discrete entropy inequalities for the coupling problem and an estimate of the discrete entropy dissipation in the proposed scheme.
Boutin B., Coquel F., and LeFloch P.G.,
Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces,
Proc. Roy. Soc. Edinburgh Sect. A, Vol. 141, 921-956 (2011).
[Abstract]
We investigate various analytical and numerical techniques for the coupling of nonlinear hyperbolic systems and, in particular, we introduce here an augmented formulation which allows for the modeling of the dynamics of interfaces between fluid flows. The main technical difficulty to be overcome lies in the possible resonance effect when wave speeds coincide and global hyperbolicity is lost. As a consequence, non-uniqueness of weak solutions is observed for the initial value problem which need to be supplemented with further admissibility conditions. This first paper is devoted to investigating these issues in the setting of self-similar vanishing viscosity approximations to the Riemann problem for general hyperbolic systems. Following earlier works by Joseph, LeFloch, and Tzavaras, we establish an existence theorem for the Riemann problem under fairly general structural assumptions on the nonlinear hyperbolic system and its regularization. Our main contribution consists of nonlinear wave interaction estimates for solutions which apply to resonant wave patterns.
Boutin B., Chalons C., and Raviart P.-A.,
Existence result for the coupling problem of two scalar conservation laws with Riemann initial data,
Math. Models Methods Appl. Sci., Vol. 20, No. 10, 1859-1898 (2010).
[Abstract]
This paper is devoted to the coupling problem of two scalar conservation laws through a fixed interface located for instance at \(x=0\). Each scalar conservation law is associated with its own (smooth) flux function an disposed on a half-space, namely \(x<0\) or \(x>0\). At interface \(x=0\) we impose a coupling condition whose objective is to enforce in a weak sense the continuity of a prescribed variable, which may differ from the conservative unknown (and the flux functions as well). We prove existence of a solution to the coupled Riemann problem using a constructive approach. The latter allows in particular to highlight interesting features like non uniqueness of both continuous and discontinuous (at interface \(x=0\)) solutions. The behavior of some numerical scheme is also investigated.
Boutin B., Chalons C., Lagoutière F., and LeFloch P.G.,
Convergent and conservative schemes for nonclassical solutions based on kinetic relations. I.,
Interfaces Free Bound., Vol. 10, No. 3, 399-421 (2008).
[Abstract]
We 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 abstract cases, including concave-convex as well as convex-concave flux-functions.

Boutin B., Coquel F., and LeFloch P.G.,
Coupling techniques for nonlinear hyperbolic equations (IV). A multidimensional finite volume framework. Submitted
[Arxiv]
Berthon C., Boutin B., Turpault R.,
Shock profiles for the Exner models.
Submitted
Boutin B., Coquel F., and LeFloch P.G.,
Coupling techniques for nonlinear hyperbolic equations (II). Resonant interfaces with internal structure.
Boutin B., Lafitte P.,
Splitting schemes for fluid-particles flows with energy exchanges.

Boutin B., Deriaz E., Hoch P., and Navaro P.,
Extension of ALE methodology to unstructured conical meshes,
Proceedings of CEMRACS 2010. ESAIM, Proc.
[Abstract]
We propose a bi-dimensional finite volume extension of a continuous ALE method on unstructured cells whose edges are parameterized by rational quadratic Bezier curves. For each edge, the control point possess a weight that permits to represent any conic (see for example [Li, Gao, and Chou, Visual Comp. 2006]) and thanks to [Guojin, and Sederberg, CADDM 1994], we are able to compute the exact area of our cells. We then give an extension of scheme for remapping step based on volume uxing [Margolin, and Shashkov, JCP 2003] and selfintersection flux [Hoch, HAL 2009]. For the rezoning phase, we propose a three step process based on moving nodes, followed by control point and weight re-adjustment. Finally, for the hydrodynamic step, we present the GLACE scheme [Carre, Del Pino, Despres, and Labourasse, JCP 2009] extension (at first-order) on conic cell using the same formalism. We only propose some preliminary first-order simulations for each steps: Remap, Pure Lagrangian and finally ALE (rezoning and remapping).
Billaud M., Boutin B., Caetano F., Faccanoni G., Lagoutière F., Kokh S., and Navoret L.,
Second order anti-diffusive Lagrange-remap scheme,
Proceedings of CEMRACS 2010. ESAIM, Proc.
[Abstract]
We build a non-dissipative second order algorithm for the approximate resolution of the one-dimensional Euler system of compressible gas dynamics with two components. The considered model was proposed in [Allaire, Clerc, and Kokh, JCP 2002]. The algorithm is based on [Kokh and Lagoutière, JCP 2010] which deals with a non-dissipative first order resolution in Lagrange-remap formalism. In the present paper we describe, in the same framework, an algorithm that is second order accurate in time and space, and that preserves sharp interfaces. Numerical results reported at the end of the paper are very encouraging, showing the interest of the second order accuracy for genuinely non-linear waves.
Boudin L., Boutin B., Fornet B., Goudon T., Lafitte P., Lagoutière F., and Merlet B.,
Fluid-particles flows: a thin spray model with energy exchanges,
Proceedings of CEMRACS 2008. ESAIM, Proceedings CEMRACS 2008 (2009).
[Abstract]
This paper is devoted to an asymptotic analysis of a fluid-particles coupled model, in the bubbling regime. On the theoretical point of view, we extend the analysis done in [Carrillo, Goudon and Lafitte, JCP 2008] for the case of an isentropic gas to the case of an ideal gas, thus adding the internal energy, or temperature, which is unknown. We formally derive the bubbling limit system in the same way as in [Carrillo, Goudon and Lafitte, JCP 2008] and propose a numerical scheme to solve this limit system.
 The numerical resolution of the non-limit system, and the numerical analysis of the asymptotic properties of the scheme (e.g. the asymptotic preserving property), as performed in [Carrillo, Goudon and Lafitte, JCP 2008], is at study.
Ambroso A., Boutin B., Coquel F., Godlewski E., and LeFloch P.G.,
Coupling two scalar conservation laws via Dafermos’ self-similar regularization,
Kunisch, Karl (ed.) et al., Numerical mathematics and advanced applications.
Proceedings of ENUMATH 2007, the 7th European conference on numerical mathematics and advanced applications, Graz, Austria, September 10--14, 2007. Berlin: Springer. 209-216 (2008).
[Abstract]
We are interested in the problem of coupling two scalar conservation laws with distinct flux-functions. This problem arises, for instance, in modeling fluid flows in media with discontinuous porosity and has important possible applications in the numerical computation of a singular pressure drop. This problem is also well-known to exhibit several technical difficulties due to the presence of nonconservative terms and to the resonant behavior of the system of equations. We present here a global approach consisting of two scalar problems in a half-space coupled through an algebraic jump relation. We view this problem as a \(2 × 2\) system of conservation laws, and introduce a viscous regularization à la Dafermos. We establish that this approximation converges as the viscosity tends to zero and we analyze the structure of the entropy solutions constructed in this way.
Boutin B., Chalons C., Lagoutière F., and LeFloch P.G.,
A sharp interface and fully conservative scheme for computing nonclassical shocks,
Kunisch, Karl (ed.) et al., Numerical mathematics and advanced applications.
Proceedings of ENUMATH 2007, the 7th European conference on numerical mathematics and advanced applications, Graz, Austria, September 10--14, 2007. Berlin: Springer. 217-224 (2008).
[Abstract]
We present a sharp interface and fully conservative numerical strategy for computing nonclassical solutions of scalar conservation laws. The difficult point is to impose at the discrete level a prescribed kinetic relation along each nonclassical discontinuity. Our method is based on a relevant reconstruction technique operating on each cell which is expected to contain a nonclassical shock. To prove the validity of our approach, we state some important stability properties and numerical abstracts are proposed. The convergence is also illustrated numerically.
Boutin B., Coquel F., and Godlewski E.,
Dafermos regularization for interface coupling of conservation laws,
Benzoni-Gavage, Sylvie (ed.) et al., Hyperbolic problems. Theory, numerics and applications.
Proceedings of the 11th international conference on hyperbolic problems, Ecole Normale Supérieure, Lyon, France, July 17--21, 2006. Berlin: Springer. 567-574 (2008).
[Abstract]
We study the coupling of two conservation laws with different fluxes at the interface \(x = 0\). The coupling condition yields (whenever possible) continuity of the solution at the interface. Thus the coupled model is not conservative in general. This gives rise to interesting questions such as nonuniqueness of self-similar solutions which we have chosen to analyze via a viscous regularization. Introducing a color function, we rewrite the problem in a conservative form involving a source term which is a Dirac measure. In turn, this leads to a nonconservative system which may be resonant. In this work, we analyze the regularization of this system by a viscous term following Dafermos’s approach. We prove the existence of a viscous solution to the Cauchy problem with Riemann data and study the convergence to the solution of the coupled Riemann problem when the viscosity parameter goes to zero.

Ambroso A., Boutin B., Chalons C., Coquel F., Galié T., Godlewski E., Lagoutière F., Raviart P.-A. and Seguin N.,
Construction de schémas de type Godunov pour les systèmes hyperboliques avec terme source,
Report R08003 of Laboratoire J.-L. Lions (2008).
Ambroso A., Boutin B., Chalons C., Coquel F., Galié T., Godlewski E., Lagoutière F., Raviart P.-A. and Seguin N.,
Couplage de modèles en thermohydraulique. Troisième rapport de synthèse,
Report R08047 of Laboratoire J.-L. Lions (2008).
Ambroso A., Boutin B., Chalons C., Coquel F., Galié T., Godlewski E., Lagoutière F., Raviart P.-A. and Seguin N.,
Model coupling in thermohydraulics. Fourth synthesis report,
Report R09057 of Laboratoire J.-L. Lions (2009).

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