Date: Feb 14, 2013 6:51 AM
Author: AMX
Subject: Re: Crank Nikolson scheme for semi linear parabolic type equation

On Wed, 13 Feb 2013 22:28:51 -0800 (PST), 
Sandeep Kumar <searchsandy1712@gmail.com> wrote:

> I am studying Cahn-Hilliard equation which is a fourth order
> parabolic type equation and wanted to implement this equation
> using numerical methods. I am trying to solve this with
> Crank-Nikolson scheme, but am not able to formulate the
> problem. The equation contains a fourth order derivative(
> square of laplacian) and also a laplacian acting on a cubing
> term. As per the scheme I have taken the mean of Ui's at n and
> n+1 time scale. After the action of laplacian on the cubic
> term, I get two terms. Now, I am not able to separate the n and
> n+1 term to get the expression into a matrix form. With my
> knowledge about CN scheme, we are supposed to separate the n
> and n+1 terms. Or, is there another way of solving the semi
> linear PDEs?
>
> The equation is: del u/ del t = - epsilonsquare*
> laplacian^2*u+ laplacian(u^3)-lapacian(u)


> boundary
> conditions Ux(0,t)=Ux(l,t)=Uxxx(0,t)=Uxxx(l,t)=0;
>


This (AFAIK) is an initial-boundary value problem. Initial with respect to
time and boundary with respect to space. CN can be used for IVP.
For BVP you have to use other method, like FDM, FEM or other.

For 1D (space) your problem can be written:

?u(x,t)/?t = - ?^2 ?^2u(x,t)/?x^2 + ?(u^3(x,t))/?x - ?u(x,t)/?x

After spatial discretisation RHS will be some expression of
discrete u(x_i,t).

Beside boundary conditions you have to specify initial condition
u(x,0) = f(x)

AMX

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