A study of upstream‐weighted high‐order differencing for approximation to flow convection

Abstract

AbstractThis paper is concerned with a number of upstream‐weighted second‐ and third‐order difference schemes. Also considered are the conventional upwind and central difference schemes for comparison. It commences with a general difference equation which unifies all the given first‐, second‐ and third‐order schemes. The various schemes are evaluated through the use of the general equation. The unboundedness and accuracy of the solutions by the difference schemes are assessed via various analyses: examination of the coefficients of the difference equation, Taylor series truncation error analysis, study of the upstream connection to numerical diffusion, single‐cell analysis. Finally, the difference schemes are tested on one‐ and two‐dimensional model problems. It is shown that the high‐order schemes suffer less from the problem of numerical diffusion than the first‐order upwind difference scheme. However, unboundedness cannot be avoided in the solutions by these schemes. Among them the linear upwind difference scheme presents the best compromise between numerical diffusion and solution unboundedness.