ANALYTICAL LINEAR NUMERICAL STABILITY CONDITIONS FOR AN ANISOTROPIC 3-DIMENSIONAL ADVECTION-DIFFUSION EQUATION

A one-timestep scheme for advective-diffusive problems in three dimensions is analysed from a numerical stability point of view. Choosing a realizable general seven-point centred discretization scheme, the amplification factor of the von Neumann method is calculated, and necessary and sufficient stability conditions for the general one-dimensional problem are retrieved. A similar analysis then leads to necessary conditions for the three-dimensional case. It is proved that the conditions obtained are also sufficient for an explicit N-dimensional case. Generalization is made to uncentered schemes and some classical results are recovered or corrected. For practical use, some miminum implicit factors necessary for stability are calculated and it is shown that the inspection of one-dimensional problems to get stability conditions can be tricky.