A modification of the invariant imbedding method for a singular boundary value problem

• Published in: Communications in nonlinear science & numerical simulation Vol. 19; no. 3; pp. 459 - 470
• Main Authors: Koshel, K.V., Ryzhov, E.A., Zyryanov, V.N.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.03.2014
• Subjects: Boundary value problems; Invariant imbedding; Matrix ordinary differential equation; Recurrence solution; Taylor–Couette flow; Invariant imbedding; Matrix ordinary differential equation; Recurrence solution; Taylor–Couette flow
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2013.07.001

•Invariant imbedding method modification of matrix boundary conditions is presented.•Algebraic transformations to regularize matrix boundary conditions are introduced.•A boundary value problem numerical solution is found using recurrence relations. We consider a dynamically-consistent analytical model of a 3D topographic vortex. The model is governed by equations derived from the classical problem of the axisymmetric Taylor–Couette flow. Using linear expansions, these equations can be reduced to a differential sixth-order equation with variable coefficients. For this differential equation, we formulate a boundary value problem, which has a number of issues for numerical solving. To avoid these issues and find the eigenvalues and eigenfunctions of the boundary value problem, we suggest a modification of the invariant imbedding method (the Riccati equation method). In this paper, we show that such a modification is necessary since the boundary conditions possess singular matrices, which sufficiently complicate the derivation of the Riccati equation. We suggest algebraic manipulations, which permit the initial problem to be reduced to a problem with regular boundary conditions. Also, we propose a method for obtaining a numerical solution of the matrix Riccati equation by means of recurrence relations, which allow us to obtain a matrizer converging to the required eigenfunction. The suggested method is tested by calculating the corresponding eigenvalues and eigenfunctions, and then, by constructing fluid particle trajectories on the basis of the eigenfunctions.