Creating nonuniform grids for FDM with multiple points of concentration

17 Views Asked by At

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?