Larger spirals in the Rock-Paper-Scissors reaction-diffusion equation
Channel: Nils Berglund
Date: 2022-03-01

Like the video https://youtu.be/bOn7m03ei0o this simulation shows a solution of a reaction-diffusion equation behaving in a similar way as the Belousov-Zhabotinsky chemical reactions, but which is easier to simulate. The diffusion coefficient D has been chosen larger than in the previous simulation (10 instead of 0.5), creating broader spirals.
 At each point in space and time, there are three concentrations u, v, and w of chemicals, which are represented by the red, blue and green components of an RGB color scheme. Denoting by rho = u + v + w the total concentration, the system of equations is given by 
d_t u = D*Delta(u) + u*(1 - rho - a*v)
d_t v = D*Delta(v) + v*(1 - rho - a*w)
d_t w = D*Delta(w) + w*(1 - rho - a*u)
where Delta denotes the Laplace operator, which performs a local average, and the parameter a is equal here to 0.75. The terms proportional to a*v, a*w and a*u denote reaction terms, in which Red is beaten by Blue, Blue is beaten be Green, and Green is beaten by Red. The situation is thus similar to the Rock-Paper-Scissors game (see https://en.wikipedia.org/wiki/Rock_paper_scissors ), and there exist simpler cellular automata with similar properties, see for instance https://softologyblog.wordpress.com/2018/03/23/rock-paper-scissors-cellular-automata
 Maciej Matyka has also made a nice video of this system, with several different values of the parameters a and D: https://youtu.be/4cJsAiL544Y

Render time: 1 hour 14 minutes
Color scheme: RGB

Music: "Puerto Rico Me Llama" by Jimmy Fontanez

See also https://images.math.cnrs.fr/Des-ondes-dans-mon-billard-partie-I.html for more explanations (in French) on  a few previous simulations of wave equations. 

The simulation solves a reaction-diffusion equation by discretization. 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html