Waves on a planet with a fractal continent скачать в хорошем качестве

Waves on a planet with a fractal continent

This is a simulation of the wave equation on a sphere, with boundary conditions given by a Julia set. The parameter of the Julia set is 0.37468 + 0.21115 i. The initial state is given by two circular waves, with opposite longitudes and positive latitude, and opposite sign. The complex plane can be conveniently projected to a sphere by stereographic projection, see for instance https://en.wikipedia.org/wiki/Riemann... . The projection used here maps the point 0 to the north pole of the sphere, and the point at infinity to the south pole, and it uses an additional scaling of the complex plane, to make the complement of the Julia set (which appears as the planet's continent) larger. 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. The fix used here is a combination of a regularization of the Laplacian at the poles and an averaging procedure around the poles. While it avoids blow-up, it does not completely prevent the north pole from deforming the wave. The video has two parts, showing the same simulation with two different color schemes: Wave height: 0:00 Average energy: 2:38 In the first part, the color hue and the radial coordinate show the wave height. In the second part, they show the energy, averaged from the beginning of the simulation. Render time: 1 hour 4 minutes Color scheme: Part 1 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric Firing Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt https://github.com/BIDS/colormap Music: "Into the Sky" by Jeremy Blake‪@RedMeansRecording‬ See also https://images.math.cnrs.fr/Des-ondes... 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... C code: https://github.com/nilsberglund-orlea... https://www.idpoisson.fr/berglund/sof... Many thanks to Marco Mancini and Julian Kauth for helping me to accelerate my code! #wave_equation #waves #fractal #Juliaset #sphere