Direct solution of the equation for the Stokes stream function

Abstract

Iterative procedures for solving the Stokes stream function equation can be quite time-consuming. This paper presents a direct procedure which can be used when the stream function vanishes on the periphery of a meridional-plane rectangle, one side of which is the axis of symmetry. The development allows for the possibility that the axial coordinate has been "stretched" in order to concentrate grid points in preselected bands. The resulting algorithm is, in fact, only marginally simpler if the coordinate is unstretched. Most steps of the development involve only classical methods of matrix algebra. The numerical technique is a matrix analog of the method of separation of variables for solving partial differential equations. One key step does employ a relatively modern technique - the QL algorithm for diagonalizing a symmetric tridiagonal matrix. Implementation of that algorithm has been comprehensively documented in the literature of the subject. © 1979.