Optimal and near - optimal advection - diffusion finite - difference schemes . Part 1 : constant coefficient 1 - D

An implicit finite difference scheme is derived which achieves exact results for the area, centroid, variance, skewness and kurtosis (flatness) of solutions to the unforced advection-diffusion equation ∂ t c + λ c + u ∂ x c − κ ∂ 2 x c = 0. Sufficient conditions for computational stability are that the grid spacing ∆x and time step ∆t be small enough that |u| ∆x κ < 12 1/2 , |u| ∆t ∆x < 1 2 1/2 At a loss of exactness in the kurtosis (flatness), the upper bound on ∆x can be removed and the upper bound on ∆t relaxed to |u| ∆t ∆x < 1. For forcing terms, the spatial moments are very accurate rather than exact.