spectral.legendre_diff_matrix#

spectral.legendre_diff_matrix(nodes: ndarray) → ndarray[source]#

Return Legendre spectral differentiation matrix at arbitrary nodes.

Constructs the spectral differentiation matrix \(D\) such that \(D\mathbf{u}\) approximates \(\frac{du}{dx}\) at the collocation nodes. The matrix is computed using Vandermonde matrices without requiring explicit quadrature.

Parameters:

nodesnp.ndarray: Collocation nodes

Returns:

np.ndarray: Differentiation matrix of shape (N, N)

Notes

The differentiation matrix is constructed as

\[D = V_x V^{-1}\]

where \(V\) is the Vandermonde matrix and \(V_x\) contains derivatives of the basis polynomials. This approach works for arbitrary node distributions.

References

Engsig-Karup, “Lecture 2: Polynomial Methods”