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Consider the diffusion of some unknown $R$ on a physically discrete, 1-D ring:

    x---------x---------x---------x    
n = 0         1         2         3

We let grid point $n=0$ be the same as grid point $n=3$, giving us our ring. We are given a diffusion coefficient $D$, which is still multiplied by some to some notion of the "Laplacian" (which in 1-D, is just the second order difference) of $R$ in order to tell us how $R$ moves diffuses between neighbouring nodes

We can use techniques such as 2nd order central differences to calculate some notion of second order difference of $R$ on these physically discrete points, if we consider the numerical discretization of the grid to be the same as the physical discretization.

What if the physical discretization is not uniform?

    x----------------x--x-------x    
n = 0                1  2       3

What notion of "second order difference" in $R$ might I now consider given that:

  1. I can't simply consider the physical discretization to be the same as the numerical discretization, without complicating things (non-uniform numerical grid spacing);
  2. but more importantly, I don't have any continuous physical situation to go back in order to derive a different second order differencing scheme using Taylor series, so I can't use some finer uniform numerical grid, which also happens to coincide with the physical points -- it would make no sense to have a numerical grid point that doesn't coincide with a physical grid point.

Is my only solution to make some sort of assumption regarding the "continuous" nature of $R$ between the grid points? Maybe I assume that $R$ varies linearly between the physically discrete points? Then the problem I outline in point 2. isn't there, and I can define a finer numerical grid? This is the solution advocated in a similar question here: How can I numerically differentiate an unevenly sampled function?

What other ideas might I consider?

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