The Orr-Sommerfeld equation on a manifold

The most effective and widely used methods for integrating the Orr-Sommerfe ld equation by shooting are the continuous orthogonalization method and the compound-matrix method. In this paper, we consider this problem from a dif ferential-geometric point of view. A new definition of orthogonalization is presented: restriction of the Orr-Sommerfeld to a complex Stiefel manifold ; and this definition leads to a new formulation of continuous orthogonaliz ation, which differs in a precise and interesting geometric way from existi ng orthogonalization routines. Present orthogonalization methods based on D avey's algorithm are shown to have a different differential-geometric inter pretation: restriction of the Orr-Sommerfeld equation to a complex Grassman ian manifold. This leads us to introduce the concept of a Grassmanian integ rator, which preserves linear independence and not necessarily orthogonalit y. Using properties of Grassmanian manifolds and their tangent spaces, a ne w Grassmanian integrator is introduced that generalizes Davey's algorithm. Furthermore, it is shown that the compound-matrix method is a dual Grassman ian integrator: it uses Plucker coordinates for integrating on a Grassmania n manifold, and this characterization suggests a new algorithm for construc ting the compound matrices. Extension of the differential-geometric framewo rk to general systems of linear ordinary differential equations is also dis cussed.