Simulate the brusselator on a random initial set. More...

Detailed Description

Simulate the brusselator on a random initial set.

Author: Mitch Richling https://www.mitchr.me

Keywords

brusselator reaction diffusion models

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Details

The Brusselator is a reaction-diffusion model for an autocatalytic chemical reaction in space and time. This system was was proposed by Ilya Romanovich Prigogine and Rene Lefever from Universite libre de Bruxelles – hence the name. The model is given by the following equations:

\[ \begin{eqnarray} \frac{\partial u_1}{\partial t} & = & 1-(b+1)u_1+au_1^2u_2+d_1 \left(\frac{\partial^2 u_1}{\partial x_1^2}+\frac{\partial^2 u_1}{\partial x_2^2}\right) \\ \frac{\partial u_2}{\partial t} & = & bu_1-au_1^2u_2+d_2 \left(\frac{\partial^2 u_2}{\partial x_1^2}+\frac{\partial^2 u_2}{\partial x_2^2}\right) \end{eqnarray} \]

The functions \( u_1 \) and \( u_2 \) represent dimensionless concentrations of two of the reactants – an activator and an inhibitor. The values \( a \) and \( b \) represent normalized reaction rates. The values \( d_1 \) and \( d_2 \) are "diffusion coefficients", and may be combined into a single value \( d=\frac{d_2}{d_1} \) like this:

\[ \begin{eqnarray} \frac{\partial u_1}{\partial t} & = & 1-(b+1)u_1+au_1^2u_2 \left(\frac{\partial^2 u_1}{\partial x_1^2}+\frac{\partial^2 u_1}{\partial x_2^2}\right) \\ \frac{\partial u_2}{\partial t} & = & bu_1-au_1^2u_2+d \left(\frac{\partial^2 u_2}{\partial x_1^2}+\frac{\partial^2 u_2}{\partial x_2^2}\right) \end{eqnarray} \]

In the code below, we use the following values for the parameters:

\[ \begin{array}{lcc} a & = & 9 \\ b & = & \frac{98}{10} \\ d & = & 3 \end{array} \]

We use a simple finite differences method to solve the equations with a step size of \( \Delta t=0.007 \) over 3000 steps. The spatial grid has a grid size of \( h=0.3 \). The spatial second derivatives at a point \( (x,y) \) are approximated with the five point stencil":

\[ \frac{\partial^2 u_j}{\partial x_1^2}+\frac{\partial^2 u_j}{\partial x_2^2} \approx \frac{u_j(x+h, y)+u_j(x-h, y)+u_j(x, y+h)+u_j(x, y-h)-4u_j(x,y)}{h^2} \]

The code uses four images to store the state of the system at two steps. At each time step it uses two of the images to update the other two. This doubles the RAM required, but simplifies the code and eliminates the need for swapping data. At the end of the run the last updated state images are combined into a 24-bit RGB image which is written to disk.

Definition in file brusselator.cpp.