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SUMMARYFor 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 thirdorder upwinding.

SUMMARYAlthough it is now well known that first-order convection schemes suffer from serious inaccuracies attributable to artificial viscosity or numerical diffusion under high-convection conditions, these methods continue to enjoy widespread popularity for numerical heat-transfer calculations, apparently owing to a perceived lack of viable high-accuracy alternatives. But alternatives are available. For example, nonoscillatory methods used in gasdynamics, including currently popular 'TVD schemes, can be easily adapted to multidimensional incompressible flow and convective transport. This, in itself, would be a major advance for numerical convective heat transfer, for example. But, as this paper shows, second-order TVD schemes form only a small, overly restrictive, subclass of a much more universal, and extremely simple, nonoscillatory flux-limiting strategy which can be applied to convection schemes of arbitrarily high-order accuracy, while requiring only a simple tridiagonal AD1 line-solver, as used in the majority of generalpurpose iterative codes for incompressible flow and numerical heat transfer. The new universal limiter and associated solution procedures form the so-called ULTRA-SHARP alternative for high-resolution nonoscillatory multidimensional steady-state high-speed convective modelling.