Frédéric Coquel

Bio: Frédéric Coquel is an academic researcher from École Polytechnique. The author has contributed to research in topics: Nonlinear system & Riemann problem. The author has an hindex of 26, co-authored 103 publications receiving 2251 citations. Previous affiliations of Frédéric Coquel include Centre national de la recherche scientifique & Pierre-and-Marie-Curie University.

Relaxation of Energy and Approximate Riemann Solvers for General Pressure Laws in Fluid Dynamics

Abstract: We consider the Euler equations for a compressible inviscid fluid with a general pressure law $p(\rho,\varepsilon)$, where $\rho$ represents the density of the fluid and $\varepsilon$ its specific internal energy. We show that it is possible to introduce a relaxation of the nonlinear pressure law introducing an energy decomposition under the form $\varepsilon= \varepsilon _1 + \varepsilon _2.$ The internal energy $\varepsilon _1$ is associated with a (simpler) pressure law $p_1(\rho,\varepsilon_1)$; the energy $\varepsilon _2$ is advected by the flow. These two energies are also subject to a relaxation process and in the limit of an infinite relaxation rate, we recover the initial pressure law p. We show that, under some conditions of subcharacteristic type, for any convex entropy associated with the pressure p, we can find a global convex and uniform entropy for the relaxation system. From our construction, we also deduce the extension to general pressure laws of classical approximate Riemann solvers for polytropic gases, which only use a single call to the pressure law (per mesh point and time step). For the Godunov scheme, we show that this extension satisfies stability, entropy, and accuracy conditions.

An error estimate for finite volume methods for multidimensional conservation laws

Abstract: In this paper, an L°°(Ll )-error estimate for a class of finite volume methods for the approximation of scalar multidimensional conservation laws is obtained. These methods can be formally high-order accurate and are defined on general triangulations. The error is proven to be of order ft'/4 , where h represents the "size" of the mesh, via an extension of Kuznetsov approximation theory for which no estimate of the total variation and of the modulus of con- tinuity in time are needed. The result is new even for the finite volume method constructed from monotone numerical flux functions.

A priori and a posteriori analysis of finite volume discretizations of Darcy's equations

Abstract: This paper is devoted to the numerical analysis of some finite volume discretizations of Darcy's equations. We propose two finite volume schemes on unstructured meshes and prove their equivalence with either conforming or nonconforming finite element discrete problems. This leads to optimal a priori error estimates. In view of mesh adaptivity, we exhibit residual type error indicators and prove estimates which allow to compare them with the error in a very accurate way.

A relaxation method for two-phase flow models with hydrodynamic closure law

Abstract: This paper is devoted to the numerical approximation of the solutions of a system of conservation laws arising in the modeling of two-phase flows in pipelines. The PDEs are closed by two highly nonlinear algebraic relations, namely, a pressure law and a hydrodynamic one. The severe nonlinearities encoded in these laws make the classical approximate Riemann solvers virtually intractable at a reasonable cost of evaluation. We propose a strategy for relaxing solely these two nonlinearities. The relaxation system we introduce is of course hyperbolic but all associated eigenfields are linearly degenerate. Such a property not only makes it trivial to solve the Riemann problem but also enables us to enforce some further stability requirements, in addition to those coming from a Chapman-Enskog analysis. The new method turns out to be fairly simple and robust while achieving desirable positivity properties on the density and the mass fractions. Extensive numerical evidences are provided.

Finite Volume Methods for Hyperbolic Problems

Abstract: Preface 1. Introduction 2. Conservation laws and differential equations 3. Characteristics and Riemann problems for linear hyperbolic equations 4. Finite-volume methods 5. Introduction to the CLAWPACK software 6. High resolution methods 7. Boundary conditions and ghost cells 8. Convergence, accuracy, and stability 9. Variable-coefficient linear equations 10. Other approaches to high resolution 11. Nonlinear scalar conservation laws 12. Finite-volume methods for nonlinear scalar conservation laws 13. Nonlinear systems of conservation laws 14. Gas dynamics and the Euler equations 15. Finite-volume methods for nonlinear systems 16. Some nonclassical hyperbolic problems 17. Source terms and balance laws 18. Multidimensional hyperbolic problems 19. Multidimensional numerical methods 20. Multidimensional scalar equations 21. Multidimensional systems 22. Elastic waves 23. Finite-volume methods on quadrilateral grids Bibliography Index.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Abstract: to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Finite Volume Methods

Abstract: Publisher Summary This chapter focuses on finite volume methods. The finite volume method is a discretization method that is well suited for the numerical simulation of various types (for instance, elliptic, parabolic, or hyperbolic) of conservation laws; it has been extensively used in several engineering fields, such as fluid mechanics, heat and mass transfer, or petroleum engineering. Some of the important features of the finite volume method are similar to those of the finite element method: it may be used on arbitrary geometries, using structured or unstructured meshes, and it leads to robust schemes. The finite volume method is locally conservative because it is based on a “balance" approach: a local balance is written on each discretization cell that is often called “control volume;” by the divergence formula, an integral formulation of the fluxes over the boundary of the control volume is then obtained. The fluxes on the boundary are discretized with respect to the discrete unknowns.