A High-Accuracy Finite-Difference Technique for Treating the Convection-Diffusion Equation
Authors: Laumbach D.D. 
Published: December 2007

Abstract
The convection-diffusion (C-D) equation arises from the conservation equations used in thermal recovery and miscible flooding. When convection predominates, this equation is very difficult to predominates, this equation is very difficult to represent numerically. The difficulty arises due to the hyperbolic character assumed as the Peclet number becomes large; consequently, the method presented here aims at providing the highest order presented here aims at providing the highest order of accuracy within this limit. The rationale underlying the treatment is to cancel a portion of the error in the convection term with that in the accumulation term. Thus, the technique presented is referred to as the truncation cancellation procedure (TCP). procedure (TCP). The application of this technique results in a new finite-difference representation of the C-D equation that is correct to the fourth order when the Peclet number approaches infinity. In this limit, when the dimensionless time step equals the dimensionless spatial increment, the discretization is exact. For very small time steps the method reduces to one previously considered to be one of the best semi-implicit discretizations of the C-D equation. For larger time steps, it yields significantly better results. The technique is applied to a linear form of the C-D equation in an equal-size grid block system; however, it is expected that the method should give favorable results in nonlinear and variable grid systems. The method is a semi-implicit one based on a three-point spatial and two-level time approximation. Thus, in one dimension, a set of difference equations is obtained that can be treated by solving a simple tridiagonal matrix. The extension of the TCP to two dimensions using a five-spatial-point star is demonstrated through an alternating-direction implicit solution method. The numerical results presented show that the TCP discretization yields presented show that the TCP discretization yields an excellent approximation in both Cartesian and radial coordinates. Comparisons with exact analytic solutions, conventional numerical techniques, and other high-accuracy numerical methods attest to the method's superiority over other formulations based on two time levels and three spatial locations.

Introduction
The purpose of this paper is to present a high-accuracy, finite-difference formulation for the convection-diffusion (C-D) equation. The discretization method presented is applied to a linear form of the C-D equation, with the expectation that the demonstrated improvements over existing methods will apply also to nonlinear forms, particularly those that are weakly nonlinear. The method is developed for equal grid spacing; however, it should give favorable results for variable grid spacing provided the grid size does not change too abruptly. provided the grid size does not change too abruptly. The difficulties associated with obtaining sufficient accuracy from conventional numerical representations of this equation were outlined quite clearly by Peaceman and Rachford. Improved methods for treating the C-D equation have been presented by Stone and Brian, Garder et al., presented by Stone and Brian, Garder et al., Price et al., Lantz, and Chaudhari. Price et al., Lantz, and Chaudhari. A significant improvement in the numerical treatment of the C-D equation was achieved by Stone and Brian. Unfortunately, the extension of this method to two dimensions has not been clear. Also, their method still possesses some oscillatory behavior in the vicinity of large gradients in the dependent variable when convection is strongly predominant This is particularly true for large time predominant This is particularly true for large time steps. The method proposed by Garder et al. uses the method of characteristics. Here the diffusion calculation is, in effect, an explicit one that consequently imposes a stability time-step limitation. Also, calculations are somewhat complicated by the moving points that must be tracked. Price et al. motivated a number of high-accuracy discretizations through the use of Galerkin's method. The discretization that they obtained through the use of chapeau basis functions is shown below to coincide with the recommended discretization of Stone and Brian. Consequently, this difference form suffers from the same oscillatory behavior in the vicinity of sharp fronts for large time steps.