The Allen-Cahn equation on the sphere, with improved behavior at the poles @NilsBerglund

Nils Berglund | The Allen-Cahn equation on the sphere, with improved behavior at the poles @NilsBerglund | Uploaded April 2024 | Updated October 2024, 6 minutes ago.
This is a remake of the video youtu.be/cPvAIkgnDbc , showing a simulation of the Allen-Cahn equation with an improved control at the poles of the sphere. The previous simulation used a rectangular simulation grid, which required some regularization of the Laplacian at the poles to avoid numerical blow-up. This however caused some slowing down near the poles. In this version, the number of points near the poles has been reduced by a blocking procedure, which allows a significant reduction of the regularization.
The video has two parts, showing the same simulation with two different representations:
3D view: 0:00
2D view: 3:10
The 2D view uses equirectangular coordinates.
The Allen-Cahn equation models systems showing phase separation, such as ferromagnets with positively and negatively magnetized domains. These appear as yellow and blue areas in the simulation, while gray areas are those without magnetization. The Allen-Cahn equation is a nonlinear variant of the heat equation. Like the heat equation, it features a term that tends to make the field constant, given by a Laplace operator performing a local average around any given point, and pushing this average towards zero. Unlike for the heat equation, however, there is another term in the equation pushing the field towards 1 if it is positive (shown here in yellow), and towards -1 if it is negative (shown here in blue).
The equation is solved by finite differences, where the Laplacian is computed in spherical coordinates. The color hue and radial coordinate show the value of the magnetization. The point of view of the observer is rotating around the polar axis of the sphere at constant latitude.

Render time: Part 1 - 1 hour 2 minutes
Part 2 - 57 minutes 5 seconds
Compression: crf 23
Color scheme: Cividis by Jamie R. Nuñez, Christopher R. Anderton, Ryan S. Renslow
journals.plos.org/plosone/article?id=10.1371/journal.pone.0199239

Music: Virtues Inherited, Vices Passed On by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 licence. creativecommons.org/licenses/by/4.0
Source: chriszabriskie.com/reappear
Artist: chriszabriskie.com

See also https://images.math.cnrs.fr/Qu-est-ce-qu-une-Equation-aux-Derivees-Partielles-Stochastique.html for more explanations (in French) on the Allen-Cahn equation.

The simulation solves the Allen-Cahn equation by discretization. The algorithm is adapted from the paper hplgit.github.io/fdm-book/doc/pub/wave/pdf/wave-4print.pdf
C code: github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html

#phase_separation #Allen-Cahn equation #spiral

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