ULTRA-SHARP solution of the Smith-Hutton problem

Highly convective scalar transport involving near-discontinuities and strong streamline curvature was addressed in a paper by Smith and Hutton in 1982, comparing several different convection schemes applied to a specially devised test problem. 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. The present paper describes an alternate strategy of using non-artificially diffusive high order methods, while maintaining strictly monotonic transitions through the use of simple flux limited 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.