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I need to solve the following partial differential equation (Cahn-Hilliard) using finite differences:

$$\frac{\partial c}{\partial t} = \nabla^2h + \cdots$$

where $h = c(1-c)(1-2c)$.

The question I want to ask is, which of

$$\nabla^2 h = \frac{h(x+h) + h(x-h) -2h(x)}{\Delta x^2}$$

or

$$\nabla^2 h = \frac{h(x+h) + h(x-h) -2h(x)}{\Delta x^2} \frac{\partial^2h}{\partial c^2},$$

is correct? (Although the second one seems dimensionally wrong.)

Also, is $\nabla^2 c^n = n(n-1)\nabla c^{n-2}$ correct?

The question is simple, but I am not able to find the answer. Any help will be appreciated.

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1 Answer 1

up vote 1 down vote accepted

The first option: $$\nabla^2 h = \frac{h(x+h) + h(x-h) -2h(x)}{\Delta x^2} = \frac{h(x+h) + h(x-h) -2h(x)}{\Delta c^2} \frac{\partial^2c}{\partial x^2},$$

Regarding the power law:

$$\nabla^2 c^n = n(n+1)c^{n-2}$$

For a proof, see here.

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