Turing-Patterns

Model from the following article.

The Strong Formulation of the problem:

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

while its weak formulation is:

$$\begin{align} \int_{\Omega} (u^{n+1}-u^n)/\Delta t\cdot v_1+\mu \nabla u^{n+1}:\nabla v_1- (au^{n+1}+bv^{n+1})v_1\thinspace d\Omega &=0 \qquad (2)\\\ \int_{\Omega} (v^{n+1}-v^n)/\Delta t\cdot v_2+\nu \nabla v^{n+1}:\nabla v_2- (cu^{n+1}+dv^{n+1})v_2\thinspace d\Omega &=0 \qquad (3) \end{align}$$

From Firedrake 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 higer resolution in /media, whereas their compressed version are:

3D-Turing-Linear-Gmres.mp4

no-plane-3D-Turing.mp4

in 2D what it looks like:

2D-Turing.mp4

Dufiet-Boissonade model for Turing's equations

The nonlinear revision of the original Turing instability reads as follows

$$\begin{cases} \frac{\partial u}{\partial t} -a(u-h) -b(v-k) -\mu \nabla^2 u =u -\alpha v^2 +\gamma uv -u^3 \\\ \frac{\partial v}{\partial t}-c(u-h)-d(v-k)-\nu \nabla^2 v =u-\beta v\qquad (4)\\ +\text{Neumann BC} \end{cases}$$

Equations in weak form:

$$\begin{align} F= & +\int_{\Omega} \frac{u_1^{n+1}}{dt}\cdot v_1\thinspace d\Omega\\\ & +\int_{\Omega} \frac{u_2^{n+1}}{dt}\cdot v_2\thinspace d\Omega\\\ & +\mu\int_{\Omega} \nabla u_1^{n+1} : \nabla v_1 \thinspace d\Omega\\\ & +\nu\int_{\Omega} \nabla u_2^{n+1} : \nabla v_2 \thinspace d\Omega\\\ & -a\int_{\Omega} \frac{u_{n_1}^{n} }{dt}\cdot v_1 \thinspace d\Omega -b\int_{\Omega} \frac{u_{n_2}^{n}}{dt}\cdot v_1 \thinspace d\Omega\\\ & -c\int_{\Omega} \frac{u_{n_1}^{n}}{dt} \cdot v_2 \thinspace d\Omega -d\int_{\Omega} \frac{u_{n_2}^{n}}{dt}\cdot v_2 \thinspace d\Omega\\\ & -\int_{\Omega} (u^{n}_{n_1}-\alpha u^n_{n_2} +\gamma u^n_{n_1}u^n_{n_2}-u^{3,n}_{n_1}) \cdot v_1 \thinspace d\Omega\\\ & -\int_{\Omega} (u^{n}_{n_1}-\beta u^{n}_{n_2}) \cdot v_2\thinspace d\Omega\\\ & + \text{Neumann BC}\\\ \end{align}$$

Solution of the nonlinear problem

Turing.Non.Linear.mp4