Second-Order Spectral Differentiation Matrices
G. E. Sneddon · SIAM Journal on Numerical Analysis · 1996
The second-order spectral differentiation matrix is required when spectral methods are used to solve second-order differential equations. For methods based on polynomials, the precise form of this matrix will depend on the method used. In each case, however, the eigenvalue with largest magnitude grows as $O(N^4 )$ which has implications for the stability when explicit methods are used to solve time-dependent problems. This paper investigates the relationship between the different matrices and shows that they are related by a rank-two update. Furthermore, it is shown that the matrix can be chosen so that the magnitude of the largest eigenvalue grows as $O(N^2 )$ rather than $O(N^4 )$, and the properties of the underlying polynomials in this case are discussed.