# Stability of Crank-Nicholson for advection diffusion equation for spatial discretization other than finite differences second-order centered

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?

• Linear stability of the Crank-Nicolson scheme (the time integration scheme) is ensured if, for the system of ODE $y'=f(y)$ ($y$ being your discrete variables and $f$ the discrete operator representing convection diffusion and others, involving the spatial schemes of any order you like) the eigenvalues of the Jacobian $\frac{df}{dy}$ all have their real parts lower than or equal to zero, i.e. if they lie within the stability domain of the temporal scheme. So computing the eigenvalues of $f$ (they may vary in time if $f$ is non-linear) will give you a good overview of the numerical stability. Apr 16 at 21:22