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.