| Abstract: | A family of positivity preserving pointwise implicit schemes applicable to source term dominated problems is constructed, where the minimum order of spatial accuracy is one and the maximum is three. It is designed for achieving steady state numerical solutions and is constructed through the analysis of appropriate model problems, where the convective fluxes for the higher‐order members are prescribed by the Chakravarthy–Osher family of total variation diminishing (TVD) schemes. Multidimensionality is facilitated by operator splitting. Numerical experimentation confirms the stability, convergence, accuracy, positivity, and computational efficiency associated with the proposed schemes. These schemes are ideally suited to solving the low‐Reynolds number turbulent k–? equations for which the positivity of k and ? and the presence of stiff source terms are critical issues. Hence, using a finite volume formulation of these schemes, the low‐Reynolds number Chien k–? turbulence model is implemented for a flat plate geometry and a series of turbulent flow (steady state) computations are carried out to demonstrate the positivity, robustness, and reliability of the algorithm. The free‐stream and initial k and ? values are specified in a very simple manner. Algorithm convergence acceleration is achieved using Multigrid techniques. The k–? model flow predictions are shown to be in agreement with empirical profiles. Copyright © 2001 John Wiley & Sons, Ltd. |
