AbstractDiscontinuous Galerkin methods have received considerable attention in recent years for problems in which advection and diffusion terms are present. Several alternatives for treating the diffusion and advective fluxes have been introduced. This report summarizes some of the methods that have been proposed.Several numerical examples are included in the paper. These present discontinuous Galerkin solutions of one‐dimensional problems with a scalar variable. Results are presented for diffusion–reaction problems and advection–diffusion problems. We discuss the performance of various formulations with respect to accuracy as well as stability of the method. Copyright © 2003 John Wiley & Sons, Ltd.