The Gray-Scott model on the sphere: Spots

Описание: This simulation of the Gray-Scott model on the sphere uses the same parameters as the simulation    • The Gray-Scott model with smaller feed rate   on the torus. The initial state consists in 8 seeds on the vertices of a cube, and has the same symmetry as the simulation grid, which is obtained my projecting the same cube on the sphere. Compared to previous simulations on the sphere, the computation of the Laplacian has been improved, as it now takes the non-constant concentration of simulation grid points into account, thereby leading to a more uniform distribution of patterns on the sphere.
The Gray-Scott model models the chemical reaction 2A + B → 3A, meaning that if two molecules of type A encounter a molecule of type B, the type B molecule is transformed into type A. In addition, type B molecules are produced at rate a (the feed rate), and type A molecules are transformed into an inert species at rate b (the kill rate).
For a large number of molecules, the system is described by the system of reaction-diffusion equations
d_t u = Delta(u) + u²v - (a+b)u
d_t v = D*Delta(v) - u²v + a(1-v)
where u and v describe respectively the concentrations of type A and type B molecules, Delta denotes the Laplace operator, and D measures the diffusion of type B molecules. The feed rate a is here equal to 0.029, while the kill rate b is equal to 0.06. The initial state is an elliptical region with only type A, surrounded by a sea with only type B.
The video has two parts, showing the same simulation with two different representation:
3D view: 0:00
2D view: 0:58
The color hue and the radial coordinate in the first part depend on the concentration of type A. In part 1, the observer turns around the sphere on a circular orbit, centered at the center of the sphere. Part 2 uses a projection in equirectangular coordinates, which means that the x-coordinate is proportional to longitude, while the y-coordinate is proportional to latitude.

This simulation is inspired by the online simulator
https://visualpde.com/sim/?preset=Gra...
that allows you to explore the effect of the different parameters on the system.

Render time: Part 1 - 1 hour 57 minutes
Part 2 - 1 hour 39 minutes
Color scheme: Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing
https://github.com/BIDS/colormap

Music: "Zen Valley" by Josh Kirsch/Media Rights Productions

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

#reaction_diffusion #Gray_Scott

The simulation solves a partial differential equation by discretization.
C code: https://github.com/nilsberglund-orlea...
https://www.idpoisson.fr/berglund/sof...