Reaction Diffusion Results

Reaction diffusion system

Informed by the ODE results, the Brusselator system is implemented as a reaction diffusion system. Using the framework from the background section, the reaction diffusion system is

where D_x and D_y are diffusion constants.

Numerical method and initial conditions

The system is solved using spectral differentiation for the diffusion, and RK4 time stepping with time step dt=0.001 for t=[0,50]. The equations are implemented on a periodic square domain, [-2.5, 2.5] x [-2.5, 2.5], with N=256 points in x and y. The code can be found in the repository in chem-oscillations/code/brusselator_reaction_diffusion/brusselator_solver.m.

The initial conditions are informed by the ODE results. From the limit cycle, we find two points out of phase from each other:

A super-gaussian hump (proportional to r⁸) is set in the center of the domain using the values at the second red line above, and the rest of the domain is intialized at the values at the first red line above. Gaussian noise with standard deviation 0.1 and mean 0 is superimposed on top.

Results

The code for these simulations is in the code/brusselator_reaction_diffusion directory.

An animation showing the concentrations X and Y is shown in the embedded video below.

Notice the center and edges remain out of phase, with the bulge slowly diffusion outward. Near the end there is evidence of spectral ringing, indicating we should be using a finer grid.

Changing the initial conditions

Inspired by the spectral ringing in the previous example, we increase the number of grid points to 512 in x and y. The initial conditions are changed to have four super-gaussians. This case shows more evidence of steady pattern formation and less spectral ringing.