Optimal and near–optimal advection–diffusion finite–difference schemes. VI Two–dimensional alternating directions

Abstract

Computationally efficient alternating–direction implicit finite–difference schemes with high accuracy are derived for the numerical solution on a regular rectangular grid of the two–dimensional constant–coefficient diffusion equation with decay, advection, off–diagonal diffusivity and sources ∂ t c + λ c + u 1 ∂ x 1 c + u 2 ∂ x 2 c - k 11 ∂ x 1 2 c - 2 k 12 ∂ x 1 ∂ x 2 c - k 22 ∂ x 2 2 c = q . The 3×3×2 compact computational module involves three consecutive spatial points in each of the perpendicular spatial directions at two successive time levels. The schemes are tested for three variants of an exactly solvable test case, corresponding to the fractionation of an initial mixture of sinking, neutral and rising particles released together in a vertically well-mixed turbulent open-channel flow with mean current not aligned with either x 1 or x 2.