Chebyshev Collocation Method for the Helmholtz Problem

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Consider the Helmholtz equation: , with the boundary conditions
and
. You can set the value of
. This Demonstration then solves this PDE using the Chebyshev collocation method adapted for 2D problems. The solution is given either as a 3D plot or a contour plot.
Contributed by: Housam Binous, Brian G. Higgins, and Ahmed Bellagi (March 2013)
Open content licensed under CC BY-NC-SA
Snapshots
Details
In the discrete Chebyshev–Gauss–Lobatto case, the interior points are given by . These points are the extrema of the Chebyshev polynomials of the first kind,
.
The Chebyshev derivative matrix at the quadrature points is an
matrix
given by
,
,
for
, and
for
and
,
where for
and
.
The discrete Laplacian is given by where
is the
identity matrix,
is the Kronecker product operator,
, and
is
without the first row and first column.
Reference
[1] L. N. Trefethen, Spectral Methods in MATLAB, Philadelphia: SIAM, 2000.
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