I am using Tchebyshev discretization to solve a system of PDEs.

Usually, I map the Tchebyshev space($\xi$, from -1 to 1) to physical space ($x$, from 0 to L) using $$x = (\xi +1)*L/2$$

Now, I also want my grids to be clustered around some point($x_c$) in the domain. Is there any standard mapping that can achieve this?

Any suggestion/ reference to standard texts are greatly appreciated.

  • $\begingroup$ Why are you interested spectral grid points close to some point? Judging from your transformation, are you by chance using a polar coordinate system? $\endgroup$ – Bort Jul 20 '17 at 9:18
  • $\begingroup$ Do you want to map from (-1, 1) to $(x_c - L/2, x_c + L/2)$ then? $\endgroup$ – nicoguaro Jul 20 '17 at 14:30
  • $\begingroup$ @Bort : My point(rather a region) of interest has a high rate of change of gradient, hence I would like to redistribute more grid points in that region. My problem is in 3-D and I use cylindrical coordinates. $\endgroup$ – Rhinocerotidae Jul 21 '17 at 4:51
  • $\begingroup$ @nicoguaro No, $x_c$ (c-cluster's center) is not the center of my domain. I want $x_c$ to be some arbitrary point inside the domain. I still want my domain to be mapped from [-1, 1] to [0, L] $\endgroup$ – Rhinocerotidae Jul 23 '17 at 14:39

In the section about adaptive Methods Chapter 16. in "Chebyshev and Fourier Spectral Methods" from John P. Boyd several different coordinate transformations together with their application in different publications are presented. I have not used any of those transformation myself but they can serve as a starting point for your problem.

y is the physical (unmapped) coordinate and x is the computational coordinate.

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I have found a simple algebraic mapping (Eq 18.12 from here) and modified it to suit my need.

The mapping is as follows:

$$x = ((1-w)\xi^3 - x_p\xi^2 + w\xi + x_p + 1 )*(L/2) $$

where $x_p = \frac{2x_c}{L}-1$; $x_c$ is approximately the point around which you want to cluster the grids;

$w\ (0 < w \ll 1)$ controls the width of the cluster.

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