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I'm familiar with finding the order of accuracy using von Neumann analysis for finite difference schemes formulated using Taylor series expansions.

But is there a similar technique for finding the order of accuracy for schemes that use a least squares fit for approximating gradients?

Currently, I am solving a 1D advection equation in advective form. I'm using Forward Euler timestepping and an upwind quadratic least squares fit with a three-point stencil. My experiments suggest an order of accuracy of ~0.7 which suggests that I've either got bugs in my implementation or the scheme is inherently slow to converge.

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  • $\begingroup$ Can you give an example of the kind of gradient you are calculating? $\endgroup$ – Bill Barth Jul 28 '15 at 16:13
  • $\begingroup$ I'm not sure how to answer, but given that I'm fitting the polynomial $\phi = ax^3 + bx^2 + cx + d$ and I set up the least squares stencil so that I always want to find $\frac{\mathrm{d} \phi(0)}{\mathrm{d}x} = c$. $\endgroup$ – hertzsprung Jul 31 '15 at 9:24

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