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I'm having two sets of grids. One is uniform and another one is not uniform. I will calculate the derivative in uniform grid points and I like to transfer(map) the derivative to the non-uniform grid points. I have the relationship between uniform and non-uniform grid points.

A similar question was already asked. I'd like to get one answer with an example. For the time sake, I consider the following simple example.

The relation between my uniform and the non-uniform grid is $$x= \zeta^2$$ Let $y(x)$ be the function for which I'd like to calculate the derivative on the non-uniform grid and $y = x^2$. So, $\frac{dy}{dx} =2x$

The uniform grid and derivative information are given, as follows:

uniform grid and derivative

The non-uniform grid: figure.

I want to find the derivative information in the non-uniform grid by mapping.

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Use the chain rule to get the derivative on the non-uniform grid, $\frac{dy}{d\zeta}$:

$$ \begin{align} \frac{dy}{d\zeta} &= \frac{dy}{dx} \frac{dx}{d\zeta}\\ &= \frac{dy}{dx} \cdot 2 \zeta \\ &= \frac{dy}{dx} \cdot 2 \sqrt{x} \\ \end{align} $$

So if you have a way of computing $\frac{dy}{dx}$ on the uniform grid, you can simply scale it by $2\sqrt{x}$ to get the corresponding $\frac{dy}{d\zeta}$ derivative.

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  • $\begingroup$ I think I didn't ask this much easy question! I want to a map function such that, that is a straight line in the second graph. Please note that the function you provided can't be a straight line! $\endgroup$ – AGN Dec 14 '18 at 16:49
  • $\begingroup$ In that case, it's not quite clear from your question what you are trying to achieve. Are you looking for some function $f(x)$ which gives a straight line when plotted against $\xi$? If so, what is the straight line you want? Tell us more about the quantities you already have, and what you are trying to calculate. $\endgroup$ – Savithru Dec 14 '18 at 18:38
  • $\begingroup$ It is quite common in computation, we solve our partial differential equations in a uniform grid (computational plane) and transform that to a non-uniform grid (physical plane). I want to know whether any existing approaches available to transform the derivative from one plane to another. $\endgroup$ – AGN Dec 15 '18 at 10:31
  • $\begingroup$ In the non-uniform plane, only grids are clustered at some point but domain remains same. I'm looking for a map that makes the one-to-one relation between uniform to non-uniform grids. Sorry for confusing that with Jacobian transformation. In CFD books, use Jacobian transformation to transfer variable. I would like to know, whether the derivative is transformable? $\endgroup$ – AGN Dec 15 '18 at 10:35
  • $\begingroup$ The method I have shown is exactly how you transfer the derivative from the uniform computational grid to the non-uniform physical grid. The Jacobian of the mapping function $x(\xi)$ definitely shows up. This is essentially the same approach that's shown in the other answer, I just substituted your particular $x(\xi)$ relationship. You can read more on this in Section 4.3 of these notes. Eq. 4.3 is consistent with what I have given. $\endgroup$ – Savithru Dec 15 '18 at 16:10

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