I need to solve the Cahn-Hilliard equation

$$\frac{\partial u}{\partial t} = \Delta(f(u) - \epsilon^2\Delta u), \hspace{.5cm}(x, t)\in \Omega\times(0, T],$$

using mixed formulation

\begin{equation}\label{mixedCH} \begin{aligned} \frac{\partial u}{\partial t} & = \Delta v, \,\ \mbox{ for} \,\ (x,t)\in\Omega\times(0, T],\\ v & = f(u) - \epsilon^2\Delta u, \,\ \mbox{ for} \,\ (x,t)\in\Omega\times(0, T],\\ u(x, 0) & = u_0(x), \,\ x\in\Omega, \end{aligned} \end{equation}

with Neumann boundary conditions for all $t\in(0, T]$

\begin{equation}\label{NBC} \frac{\partial u}{\partial n} = \frac{\partial v}{\partial n} = 0 \mbox{ on} \,\ \partial\Omega. \end{equation} in one space dimension using this \begin{align*} u_j^{n+1} - dt\Delta _hv_j^{n+1} & = u_j^{n},\\ v_j^{n+1} + \epsilon^2\Delta _hu_j^{n+1} - (u_j^{n})^2u_j^{n+1} & = -u_j^{n}, \end{align*}

The solutions are the following

figure at time step 160figure at time step 200

with $u_0(x)=\cos(\pi x/6)$ and time step $5.0000e-04$ and thickness $\epsilon = 0.0320$, mesh size $ dx = 0.0750$

I think I am getting the wrong solution.

  • $\begingroup$ It would be good to show the equation you are solving, so people can see it right here. Also, to add a few words on the numerical scheme tried - what is the method called, where it comes from etc. $\endgroup$ Sep 15 '20 at 14:26
  • $\begingroup$ What is your $f(u)$. $\endgroup$ Sep 15 '20 at 19:05
  • $\begingroup$ So it looks like we have a classical "red-black" instability here. The standard way to analyze it is assume a small perturbation $\exp(i k x - i \omega t)$ and find the linear dispersion relation for your equation and the chosen time integration algorithm. $\endgroup$ Sep 16 '20 at 3:14
  • $\begingroup$ Here,$f(u)=u^3-u.$ $\endgroup$
    – 420
    Sep 16 '20 at 7:00

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