RAIN: Robust simulation of the 2-D Cahn-Hilliard equation

Purpose

• The package "Rain" simulates the 2-D isotropic Cahn-Hilliard equation. The Legendre-Galerkin method was used for the spatial discretization, and the first- and second-order energy stabilized methods were used for the temporal discretization.

Specifications

• Name: Rain.
• Author: Feng Chen.
• Finishing date: 07/02/2011.
• Languages: Fortran 90.
• Required libraries: BLAS, LAPACK.

Simple Example

• Equation: $$\begin{equation} \begin{aligned} & \phi_t= M \Delta \mu, && \quad \text{in } \Omega, \\ & \mu = \frac{\phi(\phi^2 -1)}{\epsilon^2} - \Delta \phi, && \quad \text{in } \Omega, \\ & \partial_{\boldsymbol{n}} \phi = \partial_{\boldsymbol{n}} \mu = 0 , && \quad \text{on } \partial \Omega. \end{aligned} \end{equation}$$
• Parameters: $$\begin{equation} \Omega =(-1,1)^2, \, T=0.6, \, \delta t = 0.001, \, N_x = N_y= 128, \, M = 0.01, \, \epsilon=0.02, \, s=2. \end{equation}$$
• Exact solution and input functions: $$\begin{equation} \phi(x,y,0) = \text{two separated balls} \end{equation}$$ with radii $\frac{1}{2}$ and $\frac{1}{\sqrt{2}}$, and with centers $(-\frac12,\frac12)$ and $(\frac15,-\frac15)$. It is evolved through the first-order scheme.

Quick Start

• Compiling and running:

  cd ./Rain
  make library
  gfortran Rain_Main.cu -llibrary -llapack -lblas
  ./a.out
  

• Output:

• CPU: Intel(R) Xeon(R) CPU X5550 @2.67GHz.
• OS: CentOS release 6.4 (Final).
• Compiler: gfortran 4.5.1.

References

• Feng Chen and Jie Shen. Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications, Journal of Computational Physics, Volume 231, Issue 15, 5016-5028, (2012).
• Jie Shen and Xiaofeng Yang. Numerical Approximations of Allen-Cahn and Cahn-Hilliard Equations. DCDS, Series A, 28:1669-1691 (2010).

Code Highlights

!!Assign the initial condition for phi

case (1)! one ball.
phi(i,j) = dtanh((dsqrt(x*x+y*y)-r)/RainEps) 
case (2)! two balls.
phi(i,j) = dtanh((dsqrt((x+1d0/2d0)**2+&
     (y-1d0/2d0)**2)-1d0/4d0)/RainEps)-1d0 &
   + dtanh((dsqrt((x-1d0/5d0)**2+(y+1d0/5d0)**2)-1d0/2d0)/RainEps)-1d0
phi(i,j) = phi(i,j) + 1d0 
case (3)! three balls.
phi(i,j) = dtanh((dsqrt((x+1d0/2d0)**2+(y-1d0/2d0)**2)-&
     1d0/4d0)/RainEps)-1d0 &
   + dtanh((dsqrt((x-1d0/4d0)**2+(y+1d0/4d0)**2)-1d0/2d0)/RainEps)-1d0 &
   + dtanh((dsqrt((x+1d0/2d0)**2+(y+1d0/2d0)**2)-1d0/5d0)/RainEps)-1d0
phi(i,j) = phi(i,j) + 1d0
case (4)! two kissing balls.
phi(i,j) = dtanh((dsqrt((x+0.346d0)**2+y**2)-1d0/3d0)/RainEps)-1d0 &
   + dtanh((dsqrt((x-0.346d0)**2+y**2)-1d0/3d0)/RainEps)-1d0
phi(i,j) = phi(i,j) + 1d0 
case (5)  ! a single tilde line.
phi(i,j) = dtanh((y-2d0*x)/RainEps)-1d0  
phi(i,j) = phi(i,j) + 1d0    
case (6) ! a single verticle line.
phi(i,j) = dtanh(x/RainEps)-1d0  
phi(i,j) = phi(i,j) + 1d0