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 everywhere the von Neumann stability proof and comments about the scheme properties only for second-order centered spatial finite differences for both first and second-order derivatives (see eg here). My questions are:

1. If I use upwind, first or higher order, or higher orders centered differences, or finite elements, is the scheme always stable for the linear advection-diffusion equation? Any reference?

2. Does the unconditional stability of Crank Nicholson also holds for only advection or only diffusion, and for any physical Peclet number?

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:

1. 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?

2. Does the unconditional stability of Crank Nicholson also hold for only advection or only diffusion, and for any physical Peclet number?

Crank Nicholson mentioned 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 everywhere the von Neumann stability proof and comments about the scheme properties only for second-order centered spatial finite differences for both first and second-order derivatives (see eg here). My questions are:

1. If I use upwind, first or higher order, or higher orders centered differences, or finite elements, is the scheme always stable for the linear advection-diffusion equation? Any reference?

2. Does the unconditional stability of Crank Nicholson also holds for only advection or only diffusion, and for any physical Peclet number?

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 everywhere the von Neumann stability proof and comments about the scheme properties only for second-order centered spatial finite differences for both first and second-order derivatives (see eg here). My questions are:

1. If I use upwind, first or higher order, or higher orders centered differences, or finite elements, is the scheme always stable for the linear advection-diffusion equation? Any reference?

2. Does the unconditional stability of Crank Nicholson also holds for only advection or only diffusion, and for any physical Peclet number?