Franck Boyer

A theoretical and numerical model for the study of incompressible mixture flows

Abstract In this paper we give a complete derivation of a new model for the study of incompressible mixture flows. The equations introduced are a generalization of a model previously studied in the literature, in which the densities and the viscosities of the two phases are allowed to be different. Then, we introduce a finite-difference scheme for the numerical computations and the qualitative validation of the model. In particular, the use of an anti-diffusive second-order scheme for the transport scheme is explained and justified. One of the main physical experiment that we manage to simulate is the one of the spinodal decomposition under shear, but in order to show that the model is relevant in many general situations, we also obtain significant results in three other cases: the driven cavity, the Rayleigh–Taylor instability and the fall of a droplet.

Finite Volume Method for 2D Linear and Nonlinear Elliptic Problems with Discontinuities

In this paper we study the approximation of solutions to linear and nonlinear elliptic problems with discontinuous coefficients in the discrete duality finite volume (DDFV) framework. This family of schemes allows very general meshes and inherits the main properties of the continuous problem. In order to take into account the discontinuities and to prevent consistency defect in the scheme, we propose to modify the definition of the numerical fluxes on the edges of the mesh where the discontinuity occurs. We first illustrate our approach by the study of the 1D situation. Then, we show how to design our new scheme, called m-DDFV, and we propose its analysis. We also describe an iterative solver, whose convergence is proved, which can be used to solve the nonlinear discrete equations defining the finite volume scheme. Finally, we provide numerical results which confirm that the m-DDFV scheme significantly improves the convergence rate of the usual DDFV method for both linear and nonlinear problems.

Uniform controllability properties for space/time-discretized parabolic equations

This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use θ-schemes with

Hierarchy of consistent n-component Cahn–Hilliard systems

{\theta \in [\frac{1}2,1]}

Besov regularity and new error estimates for finite volume approximations of the p-laplacian

. For the proofs of controllability we rely on the strategy introduced by Lebeau and Robbiano (Comm Partial Differ Equ 20:335–356, 1995) for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions’ Hilbert Uniqueness Method strategy, exploiting the relaxed uniform observability estimate. Algorithms for the computation of the controls are proposed and analysed in the semi-discrete and fully-discrete cases. Additionally, we prove an error bound between the fully discrete and the semi-discrete control functions. This bound is however not uniform with respect to the space discretization. The theoretical results are illustrated through numerical experimentations.

Benchmark Proposal for the FVCA8 Conference: Finite Volume Methods for the Stokes and Navier–Stokes Equations

In arbitrary dimension, we consider the semidiscrete elliptic operator

Finite Volume Method for Nonlinear Transmission Problems

-\partial_t^2+\mathcal{A}^{\scriptscriptstyle\mathfrak{M}}

SPECTRAL ANALYSIS OF DISCRETE ELLIPTIC OPERATORS AND APPLICATIONS IN CONTROL THEORY

, where