Dante's 7th circle: a heptagonal parabolic resonator
Channel: Nils Berglund
Date: 24/10/2021

This video introduces another recently implemented color scheme: Inferno, by Nathaniel J. Smith and Stefan van der Walt --- thanks to Kram1032 for sending me the reference to a whole list of new schemes. 
Like several simulations in the "parabolic resonators" series, this one shows a wave in a cavity made of several arcs of parabola sharing the same focal point. In this case, there are 7 parabolic arcs meeting at the vertices of a regular heptagon. The colors show the energy of the waves, as indicated by the bar on the right. Inferno is good at highlighting regions with a large energy, putting less emphasis as Turbo on low energies. 
Parabolas have the property of turning a circular wave coming from their focus into a linear wave, and vice versa. For odd-sided resonators as this one, the linear waves do not hit opposite sides at the correct angle to be turned into an exact circular waves, but there is still a focusing effect, and the heptagonal symmetry should be preserved. The slow dispersion as time goes on, as well as the partial loss of the perfect symmetry, are most likely due to the simulation being made on a square grid that does not have the heptagonal symmetry, and round-off errors near the parabolic boundaries. It also highlights the fact that I did not solve the equation in only one slice of 360/7 degrees and reproduce the results. This would have been possible, but would have required some extra work for the boundary conditions, and limited the code to symmetric initial states. 

Render time: 24 minutes 40 seconds
Color scheme: Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap

Music: "Lusciousness" by Asher Fulero@AsherFulero

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 my colleague Marco Mancini for helping me to accelerate my code!