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-negative evolution 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. (C) 2003 Elsevier Science Ltd. All rights reserved.