Analysis of a fourth-order scheme for a three-dimensional convection-diffusion model problem

We derive closed form expressions for the eigenvalues and discrete solution arising from a 19-point compact discretization of a three-dimensional convection-diffusion problem. It is shown that the coefficient matrix is positive definite when the cell-Reynolds number is greater than some critical value. By analyzing the terms composing the discrete solution, we prove that an oscillation-free discrete solution is guaranteed whenever the cell-Reynolds number exceeds a value which is grid-size dependent. An interesting result is that as the mesh size is refined, this value approaches the Golden Mean.