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International audience; The paper aims to compare the accuracy and robustness of the characteristic finite element method (CFEM) and characteristic finite volume method (CFVM) for solving convection-diffusion-reaction problems on two-dimensional triangular grids. The tests are performed on a square unit domain, to which an advective field is imposed in a domain. The results show that the CFEM gives less accurate solution than CFVM for the rotation of a slotted-cylinder and rotation of Gaussian cone problems. Moreover, CFEM gives oscillate solution while the CFVM provides an oscillation-free solution for the skew flow to the grid problem.

In this paper we present an accurate stabilized FIC-FEM formulation for the 1D advection-diffusion-reaction equation in the exponential and propagation regimes using two stabilization parameters. Both the steady-state and transient solutions are considered. The stabilized formulation is based on the standard Galerkin FEM solution of the governing differential equations derived via the Finite Increment Calculus (FIC) method. The steady-state problem is considered first. The optimal value of the two stabilization parameters ensuring an exact (nodal) FEM solution using uniform meshes of linear 2-noded elements is obtained. In the absence of the absorption term the formulation simplifies to the standard one-parameter Petrov-Galerkin method for the advection-diffusion problem. For the diffusion-reaction case one stabilization parameter is just needed and the diffusion-type stabilization term is identical to that obtained by Felippa and Oñate (2007) using a variational FIC approach. A procedure for computing the stabilization parameters for the transient problem is proposed. The accuracy of the new FIC-FEM formulation is demonstrated in the solution of steady-state and transient 1D advection-diffusion-radiation problems for a the range of physical parameters and boundary conditions. Finally we outline the procedure to extend the 1D FIC-FEM formulation to multidimensions.

This paper presents a dual reciprocity boundary element method (DRBEM) formulation for the solution of steady-state convection-diffusion-reaction problems with variable velocity field at moderately high Peclet number. This scheme is based on utilising the fundamental solution of the convection-diffusion-reaction equation with constant coefficients. In this case, we decompose the velocity field into an average and a perturbation, with the latter being treated using a dual reciprocity approximation to convert the domain integrals arising in the boundary element formulation into equivalent boundary integrals. A proposed approach is implemented to treat the convective terms with variable velocity, for which the concentration is expanded as a series of functions. Four numerical experiments are included with available analytical solutions, to establish the validity of the approach and to demonstrate the efficiency of the proposed method.

We study a model linear convection–diffusion–reaction problem where both the diffusion term and the convection term are multiplied by small parameters ed and ec, respectively. Depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain. Sharp bounds for the derivatives of the solution are derived using a barrier-function technique. These bounds are applied in the analysis of a simple upwind-difference scheme on Shishkin meshes. This method is established to be almost first-order convergent, independently of the parameters ed and ec.

kaya, Adem/0000-0001-5838-3284; Sendur, Ali/0000-0001-8628-5497 WOS: 000361573200028 A numerical scheme for the convection-diffusion-reaction (CDR) problems is studied herein. We propose a finite difference method on a special grid for solving CDR problems particularly designed to treat the most interesting case of small diffusion. We use the subgrid nodes in the Link-cutting bubble (LCB) strategy [5] to construct a numerical algorithm that can easily be extended to the higher dimensions. The method adapts very well to all regimes with continuous transitions from one regime to another. We also compare the performance of the present method with the Streamline-upwind Petrov-Galerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method. (C) 2015 Elsevier Inc. All rights reserved.

The paper addresses the development of time-accurate methods for solving transient convection–diffusion–reaction problems using finite elements. Multi-stage time-stepping schemes of high accuracy are used. They are first combined with a Galerkin formulation to briefly recall the time–space discretization. Then spatial stabilization techniques are combined with high-order time-stepping schemes. Moreover, a least-squares formulation is also developed for these high-order time schemes combined with C0 finite elements (in spite of the diffusion operator and without reducing the strong form into a system of first-order differential equations). The weak forms induced by the SUPG, GLS, SGS and least-squares formulations are presented and compared. In a companion paper (Part II of this work), the phase and damping properties of the developed schemes are analysed and numerical examples are included to confirm the effectiveness of the proposed methodology for solving time-dependent convection–diffusion–reaction problems. Copyright © 2002 John Wiley & Sons, Ltd.

The 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 neighbor 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-dimension 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., Fundação para a Ciência e a Tecnologia (FCT)

The paper addresses the development of time-accurate methods for solving transient convection–diffusion –reaction problems using finite elements. The accuracy characteristics of the spatially stabilized implicit multi-stage time-stepping schemes developed in a companion paper (Part I of this work) are analysed and compared here. This is done by means of a Fourier analysis. An important improvement is observed when the order of the method is increased. Moreover, the stabilization techniques proposed (streamline-upwind Petrov–Galerkin (SUPG), Galerkin least-square (GLS), sub-grid scale (SGS) and least squares) do not degrade the phase accuracy. Finally, some examples are presented to show the applicability of these schemes. Copyright © 2002 John Wiley & Sons, Ltd.