Nils Berglund | The shallow water equation on the sphere @NilsBerglund | Uploaded July 2024 | Updated October 2024, 1 minute ago.
This is a first try at simulating a shallow water equation on a sphere. The initial state consists of two waves moving in opposite directions along the equator of the sphere. Near the end of the simulation, one can see the onset of numerical blow-up.
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space.
The equations used here include viscosity and dissipation, as described for instance in
en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , including the Coriolis force.
The video has four parts, showing the same simulation with two different color schemes and two different visualizations:
Wave height, 3D: 0:00
Velocity (norm and direction), 3D: 0:52
Wave height, 2D: 1:50
Velocity (norm and directions), 2D: 2:42
In the first and third part, the color hue and radial coordinate (in part 1) show the height of the water. In the second and fourth part, the color hue shows the direction of the flow, and the radial coordinate and luminosity show its speed (the norm of the velocity). The 2D parts use a projection in equirectangular coordinates. In the 3D parts, the point of view is slowly rotating around the sphere in a plane containing its center. The position of the sphere is represented by a line starting above the north pole, and pointing in a fixed direction.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: 3D part - 1 hour 19 minutes
2D part - 1 hour 13 minutes
Color scheme: Parts 1 and 3 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric firing
github.com/BIDS/colormap
Parts 2 and 4 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Music: "Abroad Again" by Jeremy Blake@RedMeansRecording
See also
https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the 2D shallow water equation by discretization (finite differences).
C code: 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!
#shallowwater #waves #wave
This is a first try at simulating a shallow water equation on a sphere. The initial state consists of two waves moving in opposite directions along the equator of the sphere. Near the end of the simulation, one can see the onset of numerical blow-up.
The shallow water equations are nonlinear equations, which give a better description of the motion of water than the linear wave equation. In particular, unlike the linear wave equation, they conserve the total volume of water. The linear equation gives an approximation of the solutions, when the wave height remains close to its average over space.
The equations used here include viscosity and dissipation, as described for instance in
en.wikipedia.org/wiki/Shallow_water_equations#Non-conservative_form , including the Coriolis force.
The video has four parts, showing the same simulation with two different color schemes and two different visualizations:
Wave height, 3D: 0:00
Velocity (norm and direction), 3D: 0:52
Wave height, 2D: 1:50
Velocity (norm and directions), 2D: 2:42
In the first and third part, the color hue and radial coordinate (in part 1) show the height of the water. In the second and fourth part, the color hue shows the direction of the flow, and the radial coordinate and luminosity show its speed (the norm of the velocity). The 2D parts use a projection in equirectangular coordinates. In the 3D parts, the point of view is slowly rotating around the sphere in a plane containing its center. The position of the sphere is represented by a line starting above the north pole, and pointing in a fixed direction.
The velocity field is materialized by 2000 tracer particles that are advected by the flow.
Render time: 3D part - 1 hour 19 minutes
2D part - 1 hour 13 minutes
Color scheme: Parts 1 and 3 - Viridis by Nathaniel J. Smith, Stefan van der Walt and Eric firing
github.com/BIDS/colormap
Parts 2 and 4 - Twilight by Bastian Bechtold
github.com/bastibe/twilight
Music: "Abroad Again" by Jeremy Blake@RedMeansRecording
See also
https://images.math.cnrs.fr/des-ondes-dans-mon-billard-partie-i/ for more explanations (in French) on a few previous simulations of wave equations.
The simulation solves the 2D shallow water equation by discretization (finite differences).
C code: 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!
#shallowwater #waves #wave
![[Flash warning] Two planar waves crossing a gradient index lens](https://i.ytimg.com/vi/aYvsBgyJgIE/hqdefault.jpg "[Flash warning] Two planar waves crossing a gradient index lens
The first part of this video shows some flashing due to the formation of standing waves.
This is a variant of the simulation https://youtu.be/CdpTq3dSrXA , with a different wave form. Though it looks like several parallel beams, the incoming wave is actually obtained by superimposing two planar waves, one of them angled upwards, and the other one downwards. This results in two spots forming in the focal plane of the lens.
The videos are inspired by Huygens Optics recent short https://youtube.com/shorts/VGd3Ajnp6e0 showing the principle of a gradient index lens.
Lenses focus incoming rays of light by delaying them more near the center of the lens than at its periphery. This is often done with a material of constant index of refraction, by making the lens thicker near the center, as shown for instance in the simulation https://youtu.be/rrJJBh9ubUE . However, one can also build lenses of constant thickness, by making the index of refraction of their material depend on the location in the lens. In this simulation, the index decreases like sqrt(n0² - a*r²), where r is the distance to the axis of symmetry. This results in the incoming planar waves being focused at two points in the (estimated) focal plane, marked by a vertical line. The plot to the right shows a time-averaged value of the field along that plane.
This video has two parts, showing the same evolution with two different color gradients:
Wave height: 0:00
Averaged wave energy: 1:08
In the first part, the color hue depends on the height of the wave. In the second part, it depends on the energy of the wave, averaged over a sliding time window.
There are periodic boundary conditions between the top and bottom boundaries, absorbing boundary conditions on the right boundary, and a time-periodic signal is imposed on the left boundary. The display at the right shows the signal along the focal plane, which is indicated by a vertical line.
Render time: 32 minutes 40 seconds
Compression: crf 23
Color scheme: Part 1 - Twilight by Bastian Bechtold
https://github.com/bastibe/twilight
Part 2 - Inferno by Nathaniel J. Smith and Stefan van der Walt
https://github.com/BIDS/colormap
Music: Surrender 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 Marco Mancini and Julian Kauth for helping me to accelerate my code!
#wave #lens #gradient_index")
