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In 1982, Smith and Hutton published comparative results of several different convection‐diffusion schemes applied to a specially devised test problem involving near‐discontinuities and strong streamline curvature. First‐order methods showed significant artificial diffusion, whereas higher‐order methods gave less smearing but had a tendency to overshoot and oscillate. Perhaps because unphysical oscillations are more obvious than unphysical smearing, the intervening period has seen a rise in popularity of low‐order artificially diffusive schemes, especially in the numerical heat‐transfer industry. This paper presents an alternative strategy of using non‐artificially diffusive higher‐order methods, while maintaining strictly monotonic transitions through the use of simple flux‐limiter constraints. Limited third‐order upwinding is usually found to be the most cost‐effective basic convection scheme. Tighter resolution of discontinuities can be obtained at little additional cost by using automatic adaptive stencil expansion to higher order in local regions, as needed.