Sparse and stable spectral methods

Abstract:

We construct a novel spectral method, applicable to low and high order linear ODEs with variable coefficients and general boundary conditions. The matrices of this spectral method are almost banded and have bounded condition number. The structure of the matrices allows for a scheme that grows only linearly in the degrees of freedom, achieving spectral accuracy in O(n m2) operations, where m depends only on the number of Chebyshev points needed to resolve the coefficients of the differential operator, and n on the number of Chebyshev points needed to resolve the exact solution.