Super-convergence en norme infinie du gradient pour la méthode de Shortley-Weller.

We prove in this paper the second-order super-convergence in maximum norm of the gradient for the Shortley-Weller method. Indeed, this method is known to be second-order accurate for the solution itself and for the discrete gradient, although its consistency error near the boundary is only first-order. We present a proof in the finite-difference spirit, using a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. The proof is based on a discrete Poisson equation for the discrete gradient, with second-order accurate Dirichlet boundary conditions.The advantage of this finite-difference approach is that it can provide pointwise convergence results depending on the local consistency error and the location on the computational domain.Nous présentons dans ce rapport une preuve de la super-convergence à l'ordre deux du gradient, en norme max pour la méthode de Shortley-Weller.En effet, avec cette m\'ethode le gradient discret converge à l'ordre deux même si l'erreur de troncature près du bord du domaine est d'ordre un seulement, et que la solution elle-même ne converge aussi qu'à l'ordre deux.La preuve est réalisée avec une technique de différences finies, inspirée par l'article de Ciarlet [1], et utilisant un principe du maximum discret pour obtenir des estimations des coefficients de la matrice inverse.Elle utilise une formulation discrète de l'équation de Poisson pour le gradient discret, avec des conditions au bord de Dirichlet à l'ordre deux.Cette approche par diff\'erences finies permet d'obtenir des résultats de convergence locaux, en fonction des différentes valeurs de l'erreur de troncature et de la position du point considéré sur le domaine de calcul. Elle permet aussi d'obtenir des résultats en norme max

Generating boundary conditions for a Boussinesq system

We present a new method for the numerical implementation of generating boundary conditions for a one dimensional Boussinesq system. This method is based on a reformulation of the equations and a resolution of the dispersive boundary layer that is created at the boundary when the boundary conditions are non homogeneous. This method is implemented for a simple first order finite volume scheme and validated by several numerical simulations. Contrary to the other techniques that can be found in the literature, our approach does not cause any increase in computational time with respect to periodic boundary conditions

Activation Sites Estimation using a Fast Algorithm Based on an Eikonal Equation

International audienc

A numerical method for wave-structure interactions in the Boussinesq regime

The goal of this work is to study waves interacting with partially immersed objects allowed to move freely in the vertical direction, and in a regime in which the propagation of the waves is described by the one dimensional Boussinesq-Abbott system. The problem can be reduced to a transmission problem for this Boussinesq system, in which the transmission conditions between the components of the domain at the left and at the right of the object are determined through the resolution of coupled forced ODEs in time satisfied by the vertical displacement of the object and the average discharge in the portion of the fluid located under the object. We propose a new extended formulation in which these ODEs are complemented by two other forced ODEs satisfied by the trace of the surface elevation at the contact points. The interest of this new extended formulation is that the forcing terms are easy to compute numerically and that the surface elevation at the contact points is furnished for free. Based on this formulation, we propose a second order scheme that involves a generalization of the MacCormack scheme with nonlocal flux and a source term, which is coupled to a second order Heun scheme for the ODEs. In order to validate this scheme, several explicit solutions for this wave-structure interaction problem are derived and can serve as benchmark for future codes. As a byproduct, our method provides a second order scheme for the generation of waves at the entrance of the numerical domain for the Boussinesq-Abbott system

A Cartesian Method with Second-Order Pressure Resolution for Incompressible Flows with Large Density Ratios

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY)International audienceAn Eulerian method to numerically solve incompressible bifluid problems with high density ratio is presented. This method can be considered as an improvement of the Ghost Fluid method, with the specificity of a sharp second-order numerical scheme for the spatial resolution of the discontinuous elliptic problem for the pressure. The Navier–Stokes equations are integrated in time with a fractional step method based on the Chorin scheme and discretized in space on a Cartesian mesh. The bifluid interface is implicitly represented using a level-set function. The advantage of this method is its simplicity to implement in a standard monofluid Navier–Stokes solver while being more accurate and conservative than other simple classical bifluid methods. The numerical tests highlight the improvements obtained with this sharp method compared to the reference standard first-order methods

A sharp cartesian method for the simulation of air-water interface

Abstract: We firstly present a sharp cartesian method for the simulation of incompressible flows with high density and viscosity ratios, like air-water interfaces. This method is inspired from the second-order cartesian method for elliptic problems with immersed interfaces developed i

Evaluation of the ECGI Patchwork Method Using Experimental Data in Sinus Rhythm

International audienceTorso surface and ventricular epicardial potentials were recorded simultaneously in anesthetized, closed-chest pigs (n=5) during sinus rhythm. Activation times were estimated from recorded torso potentials using three classical ECGI methods and a new method called the Patchwork Method (PM), which locally selects the optimal ECGI method and has demonstrated its efficiency with simulated data. The aim of this study was to evaluate the Patchwork method using experimental data in sinus rhythm. By comparing the classic ECGI reconstructions to recorded epicardial activation mapping, several inaccuracies in the ECGI maps are highlighted in this study. This involved inaccuracies in reconstructing activating maps, locating breakthrough sites and the production of artificial lines of block. However, the PM overcomes these restrictions, demonstrating its abilities to accurately reconstruct activation maps (CC=0.90 [0.86 ; 0.92] and RE= 0.20 [0.19 ; 0.24]) and localize epicardial breakthrough sites (LE= 17.16 [8.87 ; 22.14]). Furthermore, it reduced the frequency of artificial lines of block (2 of 5 pig hearts)

A sharp cartesian method for the simulation of air-water interface

International audienceWe firstly present a sharp cartesian method for the simulation of incompressible flows with high density and viscosity ratios, like air-water interfaces. This method is inspired from the second-order cartesian method for elliptic problems with immersed interfaces developed in [1]. Then, because a high-order interface description is necessary in this context, we present a level-set technique allowing to maintain in the course of time a third-order accuracy of the level-set itself, and thus a first order accuracy of the curvature

Modèle local de lubrification pour des écoulements Navier-Stokes incompressible de particles sphériques

The lubrication effects are short-range hydrodynamic interactions essential to thesuspension of the particles, and are usually underestimated by direct numerical simulations ofparticle laden flows. In this paper, we propose a lubrication model for a coupled volume penalizationmethod and discrete element method solver which estimates the unresolved hydrodynamic forcesand torques in an incompressible Navier-Stokes flow. Corrections are made locally on the surfaceof the interacting particles without any assumption on the global particle shape. The numericalmodel has been validated against experimental data and is shown to perform as well as existingnumerical models that are limited to spherical particles.Les forces de lubrifications jouent un rôle fondamental dans le phénomène de suspension de particules solides dans un fluide visqueux. Ces forces hydrodynamiques de courte distance d’action sont généralement partiellement capturées par la résolution numérique directe des équations de l’écoulement. L’objectif de cet article est de proposer un modèle pour corriger localement les forces et moments de lubrification. Les corrections sont faites directement sur les éléments de surfaces des particules où les effets de lubrification sont sous-estimés, sans hypothèse sur la géométrie globale des particules. Le modèle proposé a été testé sur deux cas expérimentaux dans le cas de particules sphériques