Finite difference approximations of multidimensional convection–diffusion–reaction problems with small diffusion on a special grid

Abstract

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 PetrovGalerkin (SUPG) and the Residual-Free Bubbles (RFB) methods on several benchmark problems. The numerical experiments confirm the good performance of the proposed method.

Finite difference approximations of multidimensional unsteady convection–diffusion–reaction equations

“…We have proposed a new finite difference method for multidimensional steady convection-diffusion-reaction equations which adapts very well to all regimes in [19]. There, we did a simple splitting of convection-diffusion-reaction equations into one-dimensional convection-diffusion-reaction equations.…”

“…The same splitting was done in [13]. We can consider the method [19] as an unusual extension of LCB strategy to higher dimensions. We refer to [19] for more information.…”

“…We can consider the method [19] as an unusual extension of LCB strategy to higher dimensions. We refer to [19] for more information. In this paper, we will extend this approach to multidimensional unsteady convection-diffusion-reaction equations.…”

“…We extend the method [19] which was designed for multidimensional steady convection-diffusion-reaction equations to unsteady problems. We investigate two possible different ways of combining the discretization in time and in space (where the sequence of the discretizations is interchanged).…”

“…We investigate two possible different ways of combining the discretization in time and in space (where the sequence of the discretizations is interchanged). Discretization in time is performed by using Crank-Nicolson and Backward-Euler finite difference schemes, while for the space discretization we consider the method [19]. Numerical tests are presented to show good performance of the method.…”