On weighted-mean schemes for the finite-difference approximation to the advection-diffusion equation

The weighted-mean scheme is a method for constructing finite-difference approximations of second-order partial differential equations of the advection-diffusion type using only the center and adjacent points in each space direction. The scheme tends to a centered-difference formulation for strongly diffusive cases and to an upstream formulation for strongly advective cases. The error of approximation is O(h²) or better, when h tends to zero, and the scheme assures stability and convergence to all iterative methods no matter how large the grid size. The scheme thus makes it possible to choose the biggest grid size suitable for each specific problem thereby reducing the computing time considerably.