The Eigenvalues of Second-Order Spectral Differentiation Matrices

The eigenvalues of the pseudospectral second derivative matrix with homogeneous Dirichlet boundary conditions are important in many applications of spectral methods. This paper investigates some of their properties. Numerical results show that a certain fraction of the eigenvalues approximate the eigenvalues of the continuous operator very accurately, but the errors in the remaining ones are large. It is demonstrated that the inaccurate eigenvalues correspond to those eigenfunctions of the continuous operator that cannot be resolved by polynomial interpolation in the spectral grid. In particular, it is proved that points on average per wavelength are sufficient for successful interpolation of the eigenfunctions of the continuous operator in a Chebyshev distribution of nodes, and six points per wavelength for a uniform distribution. These results are in agreement with the observed fractions of accurate eigenvalues. By using the characteristic polynomial, a bound on the spectral radius of the differentiation matrix is derived that is accurate to 2\% or better. The effect of filtering on the eigenvalues is studied numerically.