Finite difference approximations to the advection-diffusion equation

Fiadeiro and Veronis (1977) use a weighted mean scheme to develop a finite difference approximation to advection-diffusion type partial differential equations. The resulting numerical approximation tends to a centered difference formulation for strongly diffusive cases and to an upstream formulation for strongly advective cases. A simpler alternative derivation of their scheme is presented here which clarifies its basis. Using the new approach, the extension to a spatially variable grid and a similar scheme based on average cell properties are obtained. Errors associated with temporal variations are discussed and a stability criterion appropriate to forward differencing in time is presented.