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?

  • $\begingroup$ Does this help you out? scicomp.stackexchange.com/q/5402/3691 $\endgroup$
    – boyfarrell
    Apr 16 '21 at 19:34
  • $\begingroup$ That is the one I linked. It only uses centered finite difference second order. No answer to my questions (1) and (2) $\endgroup$
    – Millemila
    Apr 16 '21 at 20:15
  • $\begingroup$ Oh missed the link one first read $\endgroup$
    – boyfarrell
    Apr 16 '21 at 20:17
  • 1
    $\begingroup$ 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. $\endgroup$
    – Laurent90
    Apr 16 '21 at 21:22
  • $\begingroup$ Mmm is that a result stemming from applying von Neumann stability analysis? any reference? Without having to do the math for each scheme (re-inventing the wheel) is there in any book a table with the list of common spatial discretization schemes for which Crank Nicholson is stable? Also: what about my question 2? Is there a general result that says: if the scheme for advection-diffusion is stable, then the same scheme is stable for the diffusion equation and the advection equation separately, whichever the Peclet number? $\endgroup$
    – Millemila
    Apr 19 '21 at 13:44

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