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It's hard to tell from your phrasing and because your link is broken, but are your boundary conditions periodic or not? If the problem is periodic, then spectral methods are the way to go since Fourier series (and their discrete coutnerparts) converge much faster for periodic functions than for more general functions.
For a $1$-D problem like this, a ...
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I agree with what @davidhigh suggested: 300 gridpoints is too much.
In the 1988 paper, Fornberg said that
...the order of accuracy is generally $n-m+1$
for which 300 grid points would lead to 299th order of accuracy if I understood you correctly.
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For any time you want $$\frac{u(t,L+h) - u(t,L-h)}{2h} = K , \quad \star$$Of course $x=L+h$ is not a point in your computational domain, but from the equation above you can get an equation for $u(x, L+ h)$ and substitute in.
More precisely, if you discretize in space (with finite difference) from $x_0, \ldots, x_n$, then $x_{n+1} = L+h$ and $x_{-1} = L-h$.
...
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It is to be expected that the weights become large, see Figure 3.2-2 in Fornberg's book A Practical Guide to Pseudospectral Methods. It shows the weights of the first derivative on a uniform grid, but I expect the same trends apply also to second derivatives on non-uniform grids.
Having said that, I doubt that it makes much sense for you to make your ...
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