A Taylor-series Analysis of Finite-difference Formulations of Convection-diffusion Equations: Limitation on Grid Peclet Number and Grid Size

A simple Taylor-series analysis is made of the finite-difference formulations of the convection terms of convection-diffusion equations. A limitation is derived on grid Peclet number and grid size for a formulation to produce the expected numerical results. To verify the theory, the systematic numerical computations were carried out of a simple one-dimensional convection-diffusion equation. The results show that the grid size has stronger effects on the prediction accuracy of a formulation than the grid Peclet number does and the higher the order is of a formulation the finer the grid is required to get an acceptable physically true result. It is also demonstrated in this paper by numerical computations that the second-order-up schemes do not work better than the second-order up-winding scheme, at least for the problem solved in this paper. Therefore the second-order up-winding scheme is recommended for numerical modeling of convection-diffusion equations.