The Orr-Sommerfeld equation on a manifold

The most effective and widely used methods for integrating the Orr-Sommerfeld equation by shooting are the continuous orthogonalization method and the compound-matrix method. In this paper, we consider this problem from a differential-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 orthogonalization, which differs in a precise and interesting geometric way from existing orthogonalization routines. Present orthogonalization methods based on Davey's algorithm are shown to have a different differential-geometric interpretation: restriction of the Orr-Sommerfeld equation to a complex Grassmanian manifold. This leads us to introduce the concept of a Grassmanian integrator, which preserves linear independence and not necessarily orthogonality. Using properties of Grassmanian manifolds and their tangent spaces, a new Grassmanian integrator is introduced that generalizes Davey's algorithm. Furthermore, it is shown that the compound-matrix method is a dual Grassmanian integrator: it uses Plucker coordinates for integrating on a Grassmanian manifold, and this characterization suggests a new algorithm for constructing the compound matrices. Extension of the differential-geometric framework to general systems of linear ordinary differential equations is also discussed.