New cell–vertex reconstruction for finite volume scheme: Application to the convection–diffusion–reaction equation

Abstract

a b s t r a c tThe design of efficient, simple, and easy to code, second-order finite volume methods is an important challenge to solve practical problems in physics and in engineering where complex and very accurate techniques are not required. We propose an extension of the original Frink's approach based on a cell-to-vertex interpolation to compute vertex values with neighbour cell values. We also design a specific scheme which enables to use whatever collocation point we want in the cells to overcome the mass centre point restrictive choice. The method is proposed for two-and three-dimensional geometries and a second-order extension time-discretization is given for time-dependent equation. A large number of numerical simulations are carried out to highlight the performance of the new method.

A new energy conservation scheme for the numeric study of the heat transfer in profile extrusion calibration

“…For the Neumann condition, vertex interpolation can be preformed in two way; we introduce ghost cells where the temperature is evaluated using the Neumann condition; or we consider a stencil for the interpolation gathering cells around the the vertex. We refer to [2] for further details of the method.…”

“…Simplicity, versatility, mostly independent of the cell shape, easy to code, many practical problems in physics and engineering are now discretized with de FV method on unstructured meshes and recent progress enables to consider a wide range of applications for two or three dimensional geometries. In [2], a generic second-order finite volume scheme for the convection-diffusion-reaction equation based on the cell to vertex technology turns to be very efficient and robust and the method was experimented for non-homogeneous and anisotropic problems [3]. The specificity of this work consists in extending the method for the discontinuous situations, both for the solution and the coefficients.…”

A novel heat transfer coefficient identification methodology for the profile extrusion calibration stage

“…We provide a short description of the finite volume scheme based on the cell-to-vertex technology presented in [16], that closely follows the scheme proposed in [2]. We consider the energy equation on an open bounded polygonal domain Ω of R 2 with boundary Γ (we skip the index p for the sake of simplicity).…”

“…However, the determination of this coefficient resorting to the referred prototype faces some other difficulties since an accurate model and numerical scheme are mandatory to provide approximations leading to a correct valuation of this coefficient [15]. In [16], it is shown that the generic second-order finite volume scheme for the convection-diffusion-reaction equation based on the cell to vertex technology is very efficient and robust. In [17] the scheme is tested for non-homogeneous and anisotropic problems.…”

Finite Volume Scheme Based on Cell-Vertex Reconstructions for Anisotropic Diffusion Problems with Discontinuous Coefficients

“…We have considered the Frink-Rauch-Batina-Yang's method where a more general technique to provide the coefficients is introduced as well as the notion of target combination [4], dedicated to homogeneous and isotropic situations, i.e., the coefficients are simple constant values.…”