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Consider the Cahn-Hilliard equation

$$\frac{\partial c}{\partial t} = \nabla^2(f(c)-\varepsilon^2 \nabla^2 c)$$

defined on your favorite domain. I'm looking for a literature reference that formally derives the linear fourth-order term $-\varepsilon^2 \nabla^2(\nabla^2 c)$ as an error term/numerical viscosity term relative to a finite difference method, when trying to naively solve the nonlinear diffusion problem

$$\frac{\partial c}{\partial t} = \nabla^2(f(c)),$$

i.e. I expect the small parameter $\varepsilon^2$ to be related to the spatial and temporal stepsize parameters.

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  • $\begingroup$ I don't think this is possible: the Cahn-Hilliard equation imposes a decrease of the energy functional $E=\tfrac{1}{2}\epsilon^2(\nabla c)^2+\int fdc$, while discretization errors of the diffusion equation do not satisfy that constraint. $\endgroup$
    – Carlo Beenakker
    Commented May 27, 2022 at 6:36
  • $\begingroup$ Thanks Carlo. Is there a straightforward way to see the second part of your statement ("discretization errors of the diffusion equation do not satisfy that constraint")/ do you have a reference for this? PS, when I'm talking about relating it to "the" error, really I would be satisfied if we can show equivalence (or else prove that it can never work) "up to truncation", i.e. modulo $\mathcal{O}(\varepsilon^3)$ terms. $\endgroup$
    – ithmath
    Commented May 27, 2022 at 10:50
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    $\begingroup$ If you work out the local truncation error for a finite difference discretization based on centered differences, it's true that the leading term will involve a 4th derivative, so (at least in 1D) it will have roughly the form you want. This is done, for instance, in Chapters 2-3 of LeVeque's FD book (for just the elliptic operator) or in Chapter 9 (for the full heat equation). Take a look at section 3.5 in particular. $\endgroup$
    – David Ketcheson
    Commented May 29, 2022 at 13:29
  • $\begingroup$ Thanks David, this is exactly the sort of reference I was looking for. A 1D expansion is fine. $\endgroup$
    – ithmath
    Commented May 30, 2022 at 1:12

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