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This is a follow up to this question.

I have mapped from Chebyshev space to the physical space using the mapping $x = ((1-w)\xi^3 - x_p\xi^2 + w\xi + x_p + 1 )*(L/2)$.

When in Chebyshev space I have easily defined the spectral differentiation matrix as follows:

$$(D_N)_{00} = \frac{2N^2 +1}{6}$$ $$(D_N)_{NN} = -\frac{2N^2 +1}{6}$$ $$(D_N)_{jj} = -\frac{\xi_j}{2(1-\xi_j^2)}, \ \forall \ 1 \leq j \leq N-1$$ $$(D_N)_{ij} = \frac{c_i}{c_j} \frac{(-1)^{i+j}}{\xi_i - \xi_j}, \ \forall \ i \neq j$$

where

$ c_i= \begin{cases} 2,& \text{if } i = 0 \text{ or } N\\ 1, & \text{otherwise} \end{cases}$

$ c_j= \begin{cases} 2,& \text{if } j = 0 \text{ or } N\\ 1, & \text{otherwise} \end{cases}$

  1. How do I find the analogous differentiation matrix for the mapping
    mentioned above?
  2. Will this mapping affect the stability and convergence of collocation method?
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  • $\begingroup$ You can always choose to transform the ode / pde such the differentation matrices don't change. I can expand this to answer if you are interested. I think your transformation is already meant to be used that way. $\endgroup$ – Bort Jul 27 '17 at 12:28
  • $\begingroup$ @Bort : Please expand this as an answer $\endgroup$ – Rhinocerotidae Jul 27 '17 at 12:37
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Let's say you have derivatives of different order $u_x$ in your problem on your physical domain and a transformation $x=f(\xi)$.
Instead for solving on the original ode on the old domain, you solve the transformed ode on the new domain.

The following example is taken from Boyd's book. Be the original ode: $$a_2(x)u_{xx}+a_1(x)u_{x}+a_0(x)u=g(x)$$ you solve now: $$a_2(f(\xi))\frac{f^\prime(\xi)u_{\xi\xi}-f^{\prime\prime}(\xi)u_\xi}{f^\prime(\xi)^3}+a_1(f(\xi))\frac{u_\xi}{f^\prime(\xi)}+a_0(f(\xi))u=g(f(\xi))$$ Now you can use the defined derivative operators for $\xi$.

This is essentially a variable transformation in your derivatives: $$\frac{du}{dx}=\frac{du}{d\xi}\frac{d\xi}{dx}=\frac{du}{d\xi}\frac{1}{\frac{dx}{d\xi}}=\frac{du}{d\xi}\frac{1}{f^\prime(\xi)}$$ This can of course be expanded to the multidimensional case, although it gets a bit more complicated.

Beware that you need to transform all boundary conditions as well.

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