Laguerre collocation solutions to boundary layer type problems

Abstract

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner---Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner---Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi---Davidson type methods.