Crank Nicholson is a time discretization method (see 4th equation here). From what I see around, you can use different space discretization, such as Finite elements. But for the linear advection-diffusion equation, I see on books the von Neumann stability proof only when second-order centered spatial finite differences are used for both first and second-order derivatives (see e.g. here). My questions are:
If I use upwind, first or higher-order, higher-order centered differences, or finite elements, is the scheme always unconditionally stable for the linear advection-diffusion equation? Any reference?
Does the unconditional stability of Crank Nicholson also hold for only advection or only diffusion, and for any physical Peclet number?