Example 1: coalescence of two bubbles

Files

• Comprehensive test file: main.cpp
• Reference results for comparison: time_specialized.csv

Statement of the problem

This test corresponds to a 2D simulation of coalescence of two kissing bubbles.

The domain $\Omega$ is a square $[0,2\pi]\times[0,2\pi]$

$$\begin{align} \frac{\partial \phi}{\partial t}&= \Delta \mu \text{ in }\Omega \\[6pt] \mu &= F'(\phi) - \lambda \Delta \phi \text{ in }\Omega \end{align}$$

where $\phi$ is the phase indicator, $\mu$ the generalized chemical potential and $F'$ the derivative against $\phi$ of the potential $F$ defined by:

$$\begin{align} F(\phi)&=\frac{\phi^4}{4} - \frac{\phi^2}{2} \end{align}$$

Initial condition

The initial condition consists of two bubbles:

$$\phi = \begin{cases} 1, & \text{if } (x - \pi + 1)^2 + (y - \pi)^2 < 1 \;\; \text{or} \;\; (x - \pi - 1)^2 + (y - \pi)^2 < 1 \\ -1, & \text{otherwise} \end{cases}$$

Figure 1 : two kissing bubbles at initial state

Parameters used for the test

For this test, all parameters are equal to one except the energy gradient coefficient.

Name	Description	Symbol	Value
mob	mobility coefficient	$M_\phi$	$1.0$
lambda	energy gradient coefficient	$\lambda$	$4.10^{-4}$
omega	depth of the double-well potential	$\omega$	$1.0$

Boundary conditions

Neumann boundary conditions are prescribed on the boundary of the domain:

$$\begin{align} {\bf{n}} \cdot{} \lambda \nabla \phi&=0 \text{ on }\partial\Omega \\[6pt] {\bf{n}} \cdot{} \lambda \nabla \mu&=0 \text{ on }\partial\Omega \end{align}$$

Numerical scheme

• Time integration: Euler Implicit over the interval $t\in[0,0.5]$ (it could be extended further) with a time-step $\delta t=0.05$
• Spatial discretization: uniform grid with $N=128$ nodes in each spatial direction
• Newton solver: relative tolerance $10^{-10}$, absolute tolerance $10^{-14}$
• Iterative solver: HYPRE_GMRES
• Preconditioner: HYPRE_ILU

Results

The average value of $\phi$ is an available ouput of the simulation (see the file time_specialized.csv). It is defined by:

$$\dfrac{1}{|\Omega|}\displaystyle\int_{\Omega} \phi dv$$

For this test, the computed average value should remain constant over time.

The figure 2 shows the coalescence of the two bubbles, with a final simulation time set to $50$.

Figure 2 : coalescence two kissing bubbles