I wrote some code which basically computes the color value $c_{t+1,x,y}$ of a pixel at time $t+1$ and position $(x,y)$ by taking adding linear transformations of the values $c_{t,x,y}-c_{t-1,x,y}$, $c_{t,x+1,y}-c_{t,x,y}$ and $c_{t,x,y+1}-c_{t,x,y}$ to $c_{t,x,y}$. So I kind of solve numerically the PDE $\frac{\partial{f}}{\partial{t}}=A\frac{\partial{f}}{\partial{t}}+B\frac{\partial{f}}{\partial{x}}+C\frac{\partial{f}}{\partial{y}}$ for $3\times 3$ matrices $A,B,C$ if that makes any sense.

Here are some images, which are generated by a slight variation of this concept, working with polar coordinates, but basically what is described above. The checker pattern rotates and initializes the pixels. Also there is some Gaussian noise to avoid singularities.

I just wonder if the patterns, which can be observed in these images, are 'true', in the sense that an analytic solution to that PDE would look similar, or if they stem from numerical artefacts. Especially, I'd like to know if the patterns would change if I write a program which uses not pixels on a grid but a random Delaunay triangulation of the plane. Did anyone experiment with something similar?

  • $\begingroup$ This might be a better question for Mathematics.SE than here. The patterns you show bear some resemblance to reaction-diffusion systems, although those typically have second-order spatial derivatives, it looks like. I would guess that these equations are chaotic and thus the details of the patterns formed will be very sensitive to the underlying grid, initial conditions, etc. $\endgroup$ Aug 14, 2021 at 20:20
  • $\begingroup$ Thanks a lot for the answer! I was worried that my question is to informal for math.stackexchange.com, but I'll try nontheless! $\endgroup$
    – fweth
    Aug 14, 2021 at 20:50


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