The Chebyshev Pseudospectral Method for Third-Order Differential Equations
Abstract
Spectral methods for solving partial differential
equations have become increasingly popular because of their
ability to achieve high accuracy and cost efficiency. The
Chebyshev pseudospectral methods are applied to a typical third-
order differential equation. When the standard Chebyshev
collocation method is used, the resulting differential matrix may
generate spurious positive eigenvalues, subject to how the Neumann
boundary condition is implemented. Moreover, the differential
matrix is highly non-normal with extreme eigenvalues $O(N^6)$.
Consequently, there is a gap between Lax-stability and eigenvalue-
stability, thus the time step for stability is much below $(O^6)$.
A modified Chebyshev collocation method, which is based upon a
stretched transformation of the Chebyshev grid, is also studied.
The new derivative matrix is obtained through the chain rule.
The eigenvalues are all negative and insensitive to how the
Neumann boundary condition is implemented. Also it is near normal
with extreme eigenvalue $O(N^3)$. The finite difference method is
also applied to the same equation for the sake of comparison.
And it has instability problem due to the different stencils
applied to the boundary grid and the inside points. For
theoretical interest, the closed form of the third-order
differential Chebyshev collocation matrix is derived.
