Yohan Penel

Theoretical Study of an Abstract Bubble Vibration Model

We present the theoretical study of a hyperbolic-elliptic system of equations called Abstract Bubble Vibration (Abv) model. This simplified system is derived under non-physical assumptions from a model describing a diphasic low Mach number flow. It is thus aimed at providing mathematical properties of the coupling between the hyperbolic transport equation and the elliptic Poisson equation. We prove an existence and uniqueness result including the approximation of the time of existence for any smooth initial condition. In particular, we obtain a global-in-time existence result for small initial data. We then pay attention to properties of solutions (depending of their smoothness) such as maximum principle or evenness. In particular, an explicit formula of the mean value of solutions is given.

Coupling strategies for compressible - low Mach number flows

In order to enrich the modeling of fluid flows, we investigate in this paper a coupling between two models dedicated to distinct regimes. More precisely, we focus on the influence of the Mach number as the low Mach case is known to induce theoretical and numerical issues in a compressible framework. A moving interface is introduced to separate a compressible model (Euler with source term) and its low Mach counterpart through relevant transmission conditions. A global steady state for the coupled problem is exhibited. Numerical simulations are then performed to highlight the influence of the coupling by means of a robust numerical strategy.

A Posteriori Error Estimation for the Discrete Duality Finite Volume Discretization of the Laplace Equation

An efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the Laplace equation on very general two-dimensional meshes. The main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework. Numerical tests are performed with a stiff solution on highly nonconforming locally refined meshes and with a singular solution on triangular meshes.

Application of an AMR strategy to an abstract bubble vibration model

We describe in this paper the improvement on the numerical resolution of a fluid dynamics system by means of an Adaptive Mesh Refinement algorithm in order to handle an infinitely thin interface. This model is derived from the compressible Navier-Stokes equations in the case of diphasic flows for which both phases have a low Mach number. It consists of a coupled hyperbolic-elliptic system. The first part is numerically treated thanks to a hierarchical grid structure whereas we use the Local Defect Correction method to solve the second part.

Analysis of Apparent Topography Scheme for the Linear Wave Equation with Coriolis Force

The shallow water equations can be used to model many phenomena in geophysical fluid mechanics. For large scales, the Coriolis force plays an important role and the geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis force is an important feature. In this communication, we investigate the stability condition and the behavior of the so-called Apparent Topography scheme which is capable of capturing a discrete version of the geostrophic equilibrium.

Positivity-preserving schemes for Euler equations: Sharp and practical CFL conditions

When one solves PDEs modelling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. For instance, the underlying physical assumptions for the Euler equations are the positivity of both density and pressure variables. We consider in this paper an unstructured vertex-based tesselation in R^2. Given a MUSCL finite volume scheme and given a reconstruction method (including a limiting process), the point is to determine whether the overall scheme ensures the positivity. The present work is issued from seminal papers from Perthame and Shu (On positivity preserving finite volume schemes for Euler equations, Numer. Math. 73 (1996) 119-130) and Berthon (Robustness of MUSCL schemes for 2D unstructured meshes, J. Comput. Phys. 218 (2) (2006) 495-509). They proved in different frameworks that under assumptions on the corresponding one-dimensional numerical flux, a suitable CFL condition guarantees that density and pressure remain positive. We first analyse Berthons method by presenting the ins and outs. We then propose a more general approach adding non geometric degrees of freedom. This approach includes an optimization procedure in order to make the CFL condition explicit and as less restrictive as possible. The reconstruction method is handled independently by means of @t-limiters and of an additional damping parameter. An algorithm is provided in order to specify the adjustments to make in a preexisting code based on a certain numerical flux. Numerical simulations are carried out to prove the accuracy of the method and its ability to deal with low densities and pressures.