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If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use: $$ S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S $$ where $c=\frac{K}{5}$, and; $$ \xi_i=\sinh ^{-1}\left(\frac{-K}{c}\right)+i \Delta \xi $$ with $$ \Delta \xi=\frac{1}{N_S}\left[\sinh ^{-1}\left(\frac{S_{\max }-K}{c}\right)-\sinh ^{-1}\left(-\frac{K}{c}\right)\right] $$

The result will be more grind points around $S_i\approx K$ and lesser points everywhere else. Is there any equivalent method for multiple points of "concentration" instead of just 1? I.e. $K_1$ and $K_2$, so the grid points will be finer around $K_1$ and $K_2$ and lesser points everywhere else?

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    $\begingroup$ Let $\xi_1(i)$ be a mapping providing refined resolution at $i_1$ and $\xi_2(i)$ be a mapping providing refined resolution at $i_2$. Then $\xi_3(i)= \xi_1*\xi_2/i$ should be the mapping providing refined resolution at both $i_1$ and $i_2$; it will be a piece-wise-linear function with reduced slope at those two locations. $\endgroup$ Dec 26, 2023 at 1:11
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    $\begingroup$ @MaximUmansky I would have posted this as an answer to the question. To me, it seems to answer its key component. $\endgroup$
    – Anton Menshov
    Dec 28, 2023 at 5:22

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Let $\xi_1(𝑖)$ be a mapping providing refined resolution at $𝑖_1$ and $\xi_2(𝑖)$ be a mapping providing refined resolution at $𝑖_2$. Then $\xi_3(𝑖)=\xi_1 \xi_2/𝑖$ should be the mapping providing refined resolution at both $𝑖_1$ and $𝑖_2$; it will be a piece-wise-linear function with reduced slope at those two locations.

Here is an illustration of how this works: One mapping function (dash) is optimized for enhanced resolution at $x=0.55$, the other (dash-dot) for $x=0.3$; and the proposed combination of them (solid) has both.

enter image description here

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