How avoid square shape with Laplacian operator in reaction diffusion calculations?

Question

I have used different variants of the Laplacian operator (div grad) using 4, 8, 12, 20 and 24 of the closest points. I get problems due to the chosen coordinate system and the discretization of the Laplacian operator. See images originating from circular symmetric seeds:

It is implemented as https://www.shadertoy.com/view/3sGXWG .
The Laplacian stencils are

where red points are used in upper left in the first image,
red + green in upper middle,
red + green + blue in upper right,
red + green + blue + orange in lower left box.
red + green + blue + orange + black in lower middle
red + green in is used on lower right with whighting according to Steven Roberts answer.
The equations don't show the problem but they are
$$
frac{partial red}{partial t} = nabla^2 red (red + 4 green) \
frac{partial green}{partial t} = nabla^2 green (red + 4 green) \
red := 0.99 red + 0.01 green \
green := green + 0.05 green(1 - green) - 0.03 red - 0.001 \
    red < 0: red := max(green, 0)  \
    green < 0: green:= max(red, 0)  
$$
How can I keep spherical symmetry?

Steven Roberts · Answer

There are finite difference stencils specifically designed to have rotational symmetry.  For example, instead of the standard second order stencil
$$
frac{1}{h^2} begin{bmatrix}
& 1 & \
1 & -4 & 1\
& 1 &
end{bmatrix}
$$
you can use
$$
frac{1}{6 h^2} begin{bmatrix}
1 & 4 & 1 \
4 & -20 & 4 \
1 & 4 & 1
end{bmatrix}
$$
I first came across this in
Hundsdorfer, Willem, and Jan G. Verwer. Numerical solution of time-dependent advection-diffusion-reaction equations. Vol. 33. Springer Science & Business Media, 2013.
on page 301, but I'm sure it's in other sources as well.
When I tried this out in your shadertoy, it unfortunately didn't resolve your issue, and the pattern looked similar to the ones you provided.  I think the real issue is that the circle produced when you click is not a perfect circle, but an approximation on the finite difference grid.  This initial condition is axially symmetric but not radially symmetric.  Refining the mesh would help, but you cannot exactly represent a circle on a Cartesian grid.  My guess is that these slight approximation errors eventually cause the non-symmetric artifacts.

How avoid square shape with Laplacian operator in reaction diffusion calculations?

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