Approximating the Derivatives of a Function Using Chebyshev-Gauss-Lobatto Points

Consider the function defined by . Using the Chebyshev–Gauss–Lobatto points, it is possible to approximate the values of the two first derivatives of at these points.

This Demonstration plots , , and , as well as the error made if the first- and second-order derivatives of are approximated using Chebyshev–Gauss–Lobatto points.

As you increase the number of interior points , you can see how the error (e.g., for given by ) becomes insignificant.

In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extremums of the Chebyshev polynomial of the first kind .

The Chebyshev derivative matrix at the quadrature points , , is given by

, , for , and for , and ,

where for and .

The matrix is then used as follows: and , where is a vector formed by evaluating at , , and and are the approximations of and at the .

References

[1] P. Moin, Fundamentals of Engineering Numerical Analysis, Cambridge, UK: Cambridge University Press, 2001.

[2] S. Biringen and C-Y. Chow, An Introduction to Computational Fluid Mechanics by Example, Hoboken, NJ: Wiley, 2011.