# Creating nonuniform grids for FDM with multiple points of concentration

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?

• 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. Commented Dec 26, 2023 at 1:11
• @MaximUmansky I would have posted this as an answer to the question. To me, it seems to answer its key component. Commented Dec 28, 2023 at 5:22

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.