Implicit finite difference techniques for the advection–diffusion equation using spreadsheets Halil Karahan * Pamukkale University, Faculty of Engineering, Department of Civil Engineering, 20017 Denizli, Turkey Received 27 July 2005; received in revised form 28 December 2005; accepted 10 January 2006 Available online 28 February 2006 Abstract This study proposes one-dimensional advection–diffusion equation (ADE) with finite differences method (FDM) using implicit spreadsheet simulation (ADEISS). By changing only the values of temporal and spatial weighted parameters with ADEISS implemen- tation, solutions are implicitly obtained for the BTCS, Upwind and Crank–Nicolson schemes. The ADEISS uses iterative spreadsheet solution technique. Thus, it is not required a solution of simultaneous equations for each time step using matrix algebra. Two examples which, have the numerical and analytical solutions in literature, are solved in order to test the ADEISS performance. Both examples are solved for three schemes. It has been determined that the Crank–Nicolson scheme is in good agreement with the analytical solution; how- ever the results of the BTCS and the Upwind schemes are lower than the analytical solution. The Upwind scheme suffers from consid- erably numerical diffusion whereas the BTCS scheme does not produce numerical diffusion. Thus, it provided better results than Upwind scheme which are closer to analytical results depending on the selected parameters. The ADEISS implementation is a computationally convenient procedure for the three well-known methods in the literature: The BTCS, Upwind and Crank–Nicolson. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Advection–diffusion; Spreadsheet; Implicit finite difference techniques; Numerical diffusion 1. Introduction It is necessary to calculate the transport of fluid proper- ties or trace constituent concentrations within a fluid for many applications in water quality modeling, air pollution, meteorology, oceanography and other physical sciences. When velocity field are complex and changing in time and transport process can not be analytically calculated, so numerical approximations to the advection equation are used in many applications. The finite-difference method is a well-established numerical method which has been applied flow and transport modeling. The theories and solution techniques of the finite-difference method have been covered in a number of textbooks [1–5]. Many popular finite difference methods, such as Noye and Tan [6], used a weighted discretisation with the modi- fied equivalent partial differential equation for solving one- dimensional Advection–Diffusion Equations (ADE). Later, the authors extended this scheme to solve two-dimensional ADE [7]. The upwind scheme of Spalding [8] and the flux- corrected scheme [9] are available for the solution of the depth-averaged form of the ADE. Another widely used approach is split-operator approach [10,11], in which the advection and diffusion terms are solved by two different numerical methods. Numerical studies show that the use of central differenc- ing for the convective terms of the ADE results in negative species concentration. Lam [12] points out that the central difference approximation will overestimate the advective flux so much that if often causes a negative concentration to appear in the neighboring cells. Leonard [13] introduced an upstream interpolation method, namely QUICK (Quadratic Upstream Interpola- tion Convective Kinematics) for one-dimensional unsteady flow to prevent this situation. Later, Leonard [14] improved 0965-9978/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2006.01.003 * Tel.: +90 258 2134030; fax: +90 258 2125548. E-mail address: hkarahan@pamukkale.edu.tr www.elsevier.com/locate/advengsoft Advances in Engineering Software 37 (2006) 601–608