Simple high‐accuracy resolution program for convective modelling of discontinuities

Abstract For steady multi‐dimensional convection, the QUICK scheme has several attractive properties. However, for highly convective simulation of step profiles, QUICK produces unphysical overshoots and a few oscillations, and this may cause serious problems in non‐linear flows. Fortunately, it is possible to modify the convective flux by writing the ‘normalized’ convected control‐volume face value as a function of the normalized adjacent upstream node value, developing criteria for monotonic resolution without sacrificing formal accuracy. This results in a non‐linear functional relationship between the normalized variables, whereas standard methods are all linear in this sense. The resulting Simple High‐Accuracy Resolution Program (SHARP) can be applied to steady multi‐dimensional flows containing thin shear or mixing layers, shock waves and other frontal phenomena. This represents a significant advance in modelling highly convective flows of engineering and geophysical importance. SHARP is based on an explicit, conservative, control‐volume flux formulation, equally applicable to one‐, two‐, or three‐dimensional elliptic, parabolic, hyperbolic or mixed‐flow regimes. Results are given for the bench‐mark purely convective oblique‐step test. The monotonic SHARP solutions are compared with the diffusive first‐order results and the non‐monotonic predictions of second‐ and third‐order upwinding.