The Orr-sommerfeld Equation on a Manifold

The most eeective 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 diierential-geometric point of view. A new deenition of orthogonalization is presented: restriction of the Orr-Sommerfeld to a complex Stiefel manifold, and this deenition leads to a new formulation of continuous orthogonalization, which diiers in a precise and interesting geometric way from existing orthogonalization routines. Present orthogon-alization methods based on Davey's algorithm are shown to have a diierent diierential-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 which generalizes Davey's algorithm. Furthermore it is shown that the compound matrix method is a dual Grassmanian integrator: it uses Pl ucker coordinates for integrating on a Grassmanian manifold, and this characterization suggests a new algorithm for constructing the compound matrices. Extension of the diierential-geometric framework to general systems of linear ordinary diierential equation is also discussed.