A numerical method for solving a scalar advection-dominated transport equation with concentration-dependent sources Minghui Jin a , Roseanne M. Ford a, *, Peter T. Cummings b a Department of Chemical Engineering, University of Virginia, Charlottesville, VA, USA b Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee, TN, and Department of Chemical Engineering, University of Tennessee, Knoxville, TN, USA Received 8 July 1998; received in revised form 7 January 2002; accepted 30 December 2002 Abstract A numerical scheme, upstream biased Eulerian algorithm for transport equations with sources (UpBEATES), is developed for solving a scalar advective-dominated transport equation with concentration-independent and -dependent source terms. A control- volume method is used for spatial discretization. Time integration is invoked to yield a discrete system of integrated-flux-integrated- source form equations. The Bott’s upstream-biased Eulerian advection scheme [Moneatry and Weakly Review 117 (1989a) 1006; Moneatry and Weakly Review 117 (1989b) 2633] is employed for approximating advective fluxes. A two-level time weighting scheme is employed for the dispersive fluxes. An upstream-biased Eulerian algorithm is proposed for the concentration-dependent source term. Flux and source limiters are developed to ensure non-negativeevolution of the scalar concentration field. Numerical experiments were presented to illustrate its performance in comparison with theoretical solutions and those of conventional methods. The proposed scheme is mass-conservative, produces non-negative concentration values, exhibits low numerical dispersion, and is efficient for advection-dominated problems with concentration-dependent source terms. Like other Eulerian schemes, the Courant  /Friedrich  /Levy (CFL) stability criterion has to be met. # 2003 Elsevier Science Ltd. All rights reserved. Keywords: Bott’s scheme; Numerical dispersion; Advection-dispersion equation 1. Introduction The fate and transport of a chemical species through porous media is often subject to one or more of the following processes: chemical and biological reactions such as utilization and growth, adsorption, decay, death, external forcing such as injection and extraction (with- drawals). Mathematical description of such a problem generally leads to an advection  /dispersion equation with reactive or source terms. In many cases, the transport is advection-dominated. Numerical solution of a homogeneous scalar advec- tion equation using conventional methods (i.e. forward- in-time, central-difference-in-space scheme, and the first-order upstream scheme) is well known to be plagued with difficulties (Noorishad, Tsang, Perrochet & Musy, 1992). The difficulties are two-fold: non- physical oscillation (wiggles, ripples, over- or under- shooting) associated with the central-difference-in-space schemes, and excessive numerical dispersion associated with the first-order upstream scheme. In the absence of background concentration values, oscillation manifests itself into non-physical negative values, which should be prohibited. Excessive numerical dispersion always leads to underestimation of peak concentration values, and overestimation of extent of concentration plume, the consequence of which could be a serious one. Although the approximation of the advective term by the first-order upstream scheme is seemingly a step- down in accuracy from the second-order center differ- ence scheme, it proves to be in the right direction (Noorishad et al., 1992). Upstream biased numerical schemes, of which the first-order upstream scheme is one, are characterized by the fact that the approxima- tion of advective mass flux across an interface between two neighboring computational cells is more closely * Corresponding author. Computers and Chemical Engineering 27 (2003) 1405  /1419 www.elsevier.com/locate/compchemeng 0098-1354/03/$ - see front matter # 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0098-1354(03)00008-5