Turing-Patterns

Model from the following article.

The Strong Formulation of the problem:

$$\begin{cases} \frac{\partial X}{\partial t}=a(X-h)+b(Y-k)+\mu \nabla^2 X \\\ \frac{\partial Y}{\partial t}=c(X-h)+d(Y-k)+\nu \nabla^2 Y \qquad (1)\\ +\text{Pure NC}\\ +\text{Periodic BC} \end{cases}$$

while its weak formulation is:

$$\begin{align} \int_{\Omega} (X^{n+1}-X^n)v+\mu \Delta t \nabla X^{n+1}\nabla v- \Delta t(aX^{n+1}+bY^{n+1})v\thinspace d\Omega &=0 \qquad (2)\\\ \int_{\Omega} (Y^{n+1}-Y^n)v+\nu \Delta t \nabla Y^{n+1}\nabla v- \Delta t(cX^{n+1}+dY^{n+1})v\thinspace d\Omega &=0 \qquad (3) \end{align}$$

From FEniCS leveraging the GMRES algorithm and the ilu preconditioner $(2),(3)$ can be assembled and solved. The code can be found in /src. The outcomes I got are in /media, whereas one extract is:

no-plane-3D-Turing.mp4

in 2D what it looks like:

2D-Turing.mp4

Possibility to extend to the non-linear case

Equations in weak form:

$$\begin{align} F= & +\int_{\Omega} \frac{u_1^{n+1}\cdot v_1}{dt}\thinspace d\Omega\\\ & +\int_{\Omega} \frac{u_2^{n+1}\cdot v_2}{dt}\thinspace d\Omega\\\ & +\int_{\Omega} \nabla u_1^{n+1} : \nabla v_1 \thinspace d\Omega\\\ & +\int_{\Omega} \nabla u_2^{n+1} : \nabla v_2 \thinspace d\Omega\\\ & -\bigg(\int_{\Omega} \frac{u_{n_1}^{n} \cdot v_1}{dt}+\int_{\Omega} \frac{u_{n_2}^{n}\cdot v_2}{dt}\bigg) \thinspace d\Omega\\\ & -\int_{\Omega} (ff^{n} \cdot v_1 + gg^{n} \cdot v_2)\thinspace d\Omega =0\\\ & + \text{Pure Neuman BC} \end{align}$$