Optimal and near–optimal advection–diffusion finite–difference schemes I. Constant coefficient in one dimension

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 ∂tc + uc +u∂xc -s∂2xc = 0. Sufficient conditions for computational stability are that the grid spacing jx and time-step jt be small enough that |u|jx/s < 121/2, |u|jt/jx < 1/21/2. At a loss of exactness in the kurtosis (flatness), the upper bound on jx can be removed and the upper bound on jt relaxed to |u|jt/jx} < 1. For forcing terms, the spatial moments are very accurate rather than exact.