Beyond first‐order upwinding: The ultra‐sharp alternative for non‐oscillatory steady‐state simulation of convection

Although 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, non‐oscillatory 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, non‐oscillatory flux‐limiting strategy which can be applied to convection schemes of arbitrarily high‐order accuracy, while requiring only a simple tridiagonal ADI line‐solver, as used in the majority of general‐purpose 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 non‐oscillatory multidimensional steady‐state high‐speed convective modelling.