Chebyshev Collocation Method for the Helmholtz Problem

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.

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.