Vincent Perrier

Random integrals and correctors in homogenization

This paper concerns the homogenization of a one-dimensional elliptic equation with oscillatory random coefficients. It is well-known that the random solution to the elliptic equation converges to the solution of an effective medium elliptic equation in the limit of a vanishing correlation length in the random medium. It is also well-known that the corrector to homogenization, i.e., the difference between the random solution and the homogenized solution, converges in distribution to a Gaussian process when the correlations in the random medium are sufficiently short-range. Moreover, the limiting process may be written as a stochastic integral with respect to standard Brownian motion. We generalize the result to a large class of processes with long-range correlations. In this setting, the corrector also converges to a Gaussian random process, which has an interpretation as a stochastic integral with respect to fractional Brownian motion. Moreover, we show that the longer the range of the correlations, the larger is the amplitude of the corrector. Derivations are based on a careful analysis of random oscillatory integrals of processes with long-range correlations. We also make use of the explicit expressions for the solutions to the one-dimensional elliptic equation.

The Silurian pelagic myodocope ostracod Richteria migrans

A study of type and newly collected material of the Silurian ostracod Richteria migrans (Barrande, 1872) demonstrates that it had wide distribution, occurring in at least the Czech Republic, France, Sardinia, Wales, central Asia and probably Poland. R. migrans has biostratigraphic value, as an indicator for the mid to late part of the Ludlow Series. It was almost certainly pelagic, living predominantly in probable shelf topographic lows to marginal/off-shelf environments, characteristically with cephalopod-graptolite-bivalve-dominant associates. It had at least five to six growth stages, exhibits polymorphic variation, and its morphology provides evidence to endorse the notion that ‘entomozoaceans are myodocopes.

Diversity and Paleoenvironment of the Flora From the Nodules of the Montceau-Les-Mines Biota (Late Carboniferous, France)

Abstract The flora from the Montceau-les-Mines Lagerstätte (Massif Central, France, Late Pennsylvanian) is preserved in small sideritic concretions and was studied in three locations in the (1) Saint-Louis, (2) Saint-François, and (3) Sainte-Hélène opencast mines. Qualitative and quantitative analyses of plant diversity and floristic composition in 6812 nodules indicate substantial variations in the floral composition of these opencast mines. More than 50 taxa are recognized and belong to groups typical of the Late Pennsylvanian flora (lycopsids, sphenopsids, tree ferns, and pteridosperms). Arborescent sphenopsids and tree ferns were the major components at Saint-Louis, whereas the flora from Saint-François consisted mainly of pteridosperms; the one from Sainte-Hélène has a more balanced composition. Taphonomic and sedimentological data show that the flora contained in the nodules was hypoautochthonous to parautochthonous. The Montceau Basin displayed a mosaic of paleoenvironments (e.g., deltaic lacustrine, paludal to fluvial) which favored colonization by plants and animals.

Asymptotic Expansion of a Multiscale Numerical Scheme for Compressible Multiphase Flow

The simulation of compressible multiphase problems is a difficult task for modelization and mathematical reasons. Here, thanks to a probabilistic multiscale interpretation of multiphase flows, we construct a numerical scheme that provides a solution to these difficulties. Three types of terms can be identified in the scheme in addition to the temporal term. One is a conservative term, the second one plays the role of a nonconservative term that is related to interfacial quantities, and the last one is a relaxation term that is associated with acoustic phenomena. The key feature of the scheme is that it is locally conservative, contrarily to many other schemes devoted to compressible multiphase problems. In many physical situations, it is reasonable to assume that the relaxation is instantaneous. We present an asymptotic expansion of the scheme that keeps the local conservation properties of the original scheme. The asymptotic expansion relies on the understanding of an equilibrium variety. Its structure depends, in principle, on the Riemann solver. We show that it is not the case for several standard solvers, and hence this variety is characterized by the local pressure and velocity of the flow. Several numerical test cases are presented in order to demonstrate the potential of this technique.

Runge-Kutta discontinuous Galerkin method for the approximation of Baer and Nunziato type multiphase models

A high-order numerical method is developed for the computation of compressible multiphase flows. The model we use is based on the Baer and Nunziato type systems [4]. Among all the other available models in the literature, these systems present the advantage to be able to simulate either interface or mixture problems. Nevertheless, they still raise some issues, mainly based on their non-conservative feature. The numerical method we propose is a discontinuous Galerkin type. In this work, the interior side integrals are computed thanks to [2]. Robustness and high order of accuracy of the method are proved on classical interface problems, but also on suitably derived analytical solutions.

The Chapman–Jouguet Closure for the Riemann Problem with Vaporization

This work is devoted to the modeling of phase transition. The thermodynamic model for phase transition chosen is a model with two equations of state, each of them modeling one phase of a given fluid. The mixture equation of state is obtained by an entropy optimization criterion. Both equations of state are supposed to be convex, and a necessary condition is found to ensure the convexity of the mixture equation of state. Then we investigate the Riemann problem for the Euler system with these equations of state. More precisely, we propose to take into account metastable states, which may occur as noted in [J. R. Simoes-Moreira and J. E. Shepherd, J. Fluid Mech., 382 (1999), pp.63–86]. We check whether the Chapman–Jouguet theory can be applied in our context, and that it is consistent with the entropy growth criterion. As the characteristic Lax criterion does not hold for this solution, an additional relation, the kinetic closure, is necessary. The common closure, i.e., the Chapman–Jouguet closure, is proved...

A Conservative Method for the Simulation of the Isothermal Euler System with the van-der-Waals Equation of State

In this article, we are interested in the simulation of phase transition in compressible flows, with the isothermal Euler system, closed by the van-der-Waals model. We formulate the problem as an hyperbolic system, with a source term located at the interface between liquid and vapour. The numerical scheme is based on (Abgrall and Saurel, J.xa0Comput. Phys. 186(2):361–396, 2003; Lexa0Métayer et al., J. Comput. Phys. 205(2):567–610, 2005). Compared with previous discretizations of the van-der-Waals system, the novelty of this algorithm is that it is fully conservative. Its Godunov-type formulation allows an easy implementation on multi-dimensional unstructured meshes.

Godunov-type schemes with an inertia term for unsteady full Mach number range flow calculations

An inertia term is introduced in the AUSM+-up scheme. The resulting scheme, called AUSM-IT (IT for Inertia Term), is designed as an extension of the AUSM+-up scheme allowing for full Mach number range calculations of unsteady flows including acoustic features. In line with the continuous asymptotic analysis, the AUSM-IT scheme satisfies the conservation of the discrete linear acoustic energy at first order in the low Mach number limit. Its capability to properly handle low Mach number unsteady flows, that may include acoustic waves or discontinuities, is numerically illustrated. The approach for building the AUSM-IT scheme from the AUSM+-up scheme is applicable to any other Godunov-type scheme.

High Order Nonlinear Numerical Schemes for Evolutionary PDEs

This book collects papers presented during the European Workshop on High Order Nonlinear Numerical Methods for Evolutionary PDEs (HONOM 2013) that was held at INRIA Bordeaux Sud-Ouest, Talence, France in March, 2013. The central topic is high order methods for compressible fluid dynamics. In the workshop, and in this proceedings, greater emphasis is placed on the numerical than the theoretical aspects of this scientific field. The range of topics is broad, extending through algorithm design, accuracy, large scale computing, complex geometries, discontinuous Galerkin, finite element methods, Lagrangian hydrodynamics, finite difference methods and applications and uncertainty quantification. These techniques find practical applications in such fields as fluid mechanics, magnetohydrodynamics, nonlinear solid mechanics, and others for which genuinely nonlinear methods are needed.

Runge–Kutta Discontinuous Galerkin Method for Multi–phase Compressible Flows

The aim of this contribution is to develop a high order numerical scheme for simulating compressible multiphase flows. For reaching high order, we propose to use the Runge-Kutta Discontinuous Galerkin method. The development of such a method is not straightforward, because it was originally developed for conservative systems, whereas the system of interest is not conservative. We show how to circumvent this difficulty, and prove the accuracy and the robustness of our method on one and two dimensional numerical tests.