Spectral integration of linear boundary value problems

Spectral integration was deployed by Orszag and co-workers (1977, 1980, 1981) to obtain stable and efficient solvers for the incompressible Navier–Stokes equation in rectangular geometries. Since then several variations of spectral integration have appeared in the literature. In this article, we derive yet more versions of spectral integration. These new versions of spectral integration rely exclusively on banded matrices as opposed to banded matrices bordered with dense rows. In addition, we derive a factored form of spectral integration which relies only on bi- and tri-diagonal matrices. Key properties, such as the accuracy of spectral integration even when Green’s functions are not resolved by the underlying grid and the accuracy of spectral integration in spite of ill-conditioning of underlying linear systems are investigated. Timed comparisons show that reducing spectral integration to bi- and tri-diagonal systems leads to significant speed-ups.