I'm implementing the Chebyshev collocation method for solving PDEs, more specifically the shallow water equations. I know the Chebyshev differentiation matrices (or differential operators) are, for default, for functions defined on the domain [-1,+1], which is a characteristic of the Chebyshev polynomials. My question is: if I want to evaluate the derivatives on an arbitrary domain [a,b], which alterations should I carry on the Chebyshev differentiation matrices? I looked out for this on the Trefethen's book "Spectral Methods in Matlab", but haven't found any mentions to this issue. Thanks in advance.
Asked
Active
Viewed 671 times
2
-
Page 21 of this pdf discusses Chebyshev Differentiation Matrices on $[a,b]$, but deals with this by immediately rescalling the boundary value problem from $[a,b]$ to $[-1,1]$. – Mark Schultz-Wu Jul 16 '16 at 00:11
2 Answers
2
Page 98 of the book you mention, Trefethen's "Spectral Methods in Matlab", gives the following formulas for the interval $[-L, L]$:
$x_L = xL$,
$D_L = D/L$,
On the same page you see how to proceed if a higher order derivative is needed, rescale $D$ first, then compute the higher order matrix, for example
$D^2_L = (D/L)^2$.
milez
- 235
1
I currently use the Chebyshev differentiation matrices and the Weideman & Reddy Matlab package to obtain them.
In their paper, http://dl.acm.org/citation.cfm?doid=365723.365727, they rescale the domain $[a,b]$ to $[-1,+1]$ with a simple linear transformation.
I would suggest you to do the same to your data, apply any differentiation, and then scale back to your original domain.
Geoffroy
- 21