Energy flux of waves with dodecahedral symmetry
Channel: Nils Berglund
Date: 21/08/2023

This simulation of waves on a sphere with dodecahedral symmetry uses the same initial state as in the recent video https://youtu.be/tSzsrCHc3xc , but the color gradient shows the energy flux, instead of the wave height and average energy. Since I could not decide which cyclic color gradient I preferred, HSL/Jet or Twilight, I used both in this video.
The wave equation is simulated on a sphere, where circular obstacles have been placed on the vertices of a regular icosahedron. The initial state consists in 12 sources, localized at the centers of the faces of the dodecahedron, which form a regular icosahedron. The point of view rotates around the sphere in the course of the simulation.
Simulating the wave equation on a sphere is rather similar to the planar case, except that the Laplacian is written in spherical coordinates. A difficulty is that the Laplacian is singular at the poles of the sphere, which induces some numerical instability. This problem is however avoided here, thanks to the boundary conditions.
The video has two parts, showing the same simulation with two different color schemes:
HSL/Jet: 0:00
Twilight: 1:52
In both parts, the radial coordinate depends on the norm of the energy flux of the wave, while the hue and luminosity depend on its direction. 

Render time: 53 minutes 28 seconds
Color scheme: Part 1 - HSL/Jet
Part 2 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight

Music:  Rubix Cube by Audionautix is licensed under a Creative Commons Attribution 4.0 licence. https://creativecommons.org/licenses/by/4.0/
Artist: http://audionautix.com/

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 the wave equation by discretization. The algorithm is adapted from the paper https://hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf 
C code: https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code!

#wave_equation #waves #sphere #dodecahedron