Approximating the Derivatives of a Function Using Chebyshev-Gauss-Lobatto Points
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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.
Contributed by: Housam Binous, Brian G. Higgins, and Ahmed Bellagi (March 2013)
Open content licensed under CC BY-NC-SA
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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.
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