Semi-Lagrangian methods have traditionally been developed in the framework of
hyperbolic equations, but several extensions of the Semi-Lagrangian approach to
diffusion and advection--diffusion problems have been proposed recently. These
extensions are mostly based on probabilistic arguments and share the common
feature of treating second-order operators in trace form, which makes them
unsuitable for mass conservative models like the classical formulations of
turbulent diffusion employed in computational fluid dynamics. We propose here
some basic ideas for treating second-order operators in divergence form. A
general framework for constructing consistent schemes in one space dimension is
presented, and a specific case of nonconservative discretization is discussed
in detail and analysed. Finally, an extension to (possibly nonlinear) problems
in an arbitrary number of dimensions is proposed. Although the resulting
discretization approach is only of first order in time, numerical results in a
number of test cases highlight the advantages of these methods for applications
to computational fluid dynamics and their superiority over to more standard low
order time discretization approaches