A lattice Boltzmann equation for diffusion

The formulation of lattice gas automata (LGA) for given partial differential equations is not straightforward and still requires some sort of magic. Lattice Boltzmann equation (LBE) models are much more flexible than LGA because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements. Here a LBE is presented for diffusion in an arbitrary number of dimensions. The model is probably the simplest LBE which can be formulated. It is shown that the resulting algorithm with relaxation parameter =1 is identical to an explicit finite-difference (EFD) formulation at its stability limit. Underrelaxation (0<<1) allows stable integration beyond the stability limit of EFD. The time step of the explicit LBE integration is limited by accuracy and not by stability requirements.