Mapping derivative information in uniform to non-uniform grid

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:

The non-uniform grid: .

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

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.
• 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. – Savithru Dec 14 '18 at 18:38
• 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. – Savithru Dec 15 '18 at 16:10