I'd guess that these patterns are supposed to look perfectly circular but they look square-like in this slide. Is this due to aliasing since the patterns have pretty low resolution on a square grid, or is it a more intrinsic property of our approximation of the PDE. The discrete Laplacian stencil that we're using looks like its trying to approximate a circle on a square grid but its not that great with just 5 points. Would the pattern look more circular if we used a 9 point Laplacian or some other more complex approximation?
keenan
@0x484884 Yes, excellent point. Just because you have a consistent approximation of the differential operator (the Laplacian, in this case) does not mean that you get a consistent approximation of the solution to the PDE (very loosely speaking, this is something like Gamma convergence).
In terms of getting waves that look "visually circular," you're right that using a higher-order Laplacian (like the 9-point version you mention) will do better.
I'd guess that these patterns are supposed to look perfectly circular but they look square-like in this slide. Is this due to aliasing since the patterns have pretty low resolution on a square grid, or is it a more intrinsic property of our approximation of the PDE. The discrete Laplacian stencil that we're using looks like its trying to approximate a circle on a square grid but its not that great with just 5 points. Would the pattern look more circular if we used a 9 point Laplacian or some other more complex approximation?
@0x484884 Yes, excellent point. Just because you have a consistent approximation of the differential operator (the Laplacian, in this case) does not mean that you get a consistent approximation of the solution to the PDE (very loosely speaking, this is something like Gamma convergence).
In terms of getting waves that look "visually circular," you're right that using a higher-order Laplacian (like the 9-point version you mention) will do better.