A new algorithm for numerical path following applied to an example from hydrodynamical flow. (English) Zbl 0724.65051
The authors present a new algorithm for numerically following solution paths and predicting singular points of large and sparse one-parameter problems \(F(x,\lambda)=0,\) where F: \({\mathbb{R}}^ n\times {\mathbb{R}}\to {\mathbb{R}}^ n\). The path following algorithm is the Euler-Newton continuation method. The involved linear systems are solved iteratively by a preconditioned Arnoldi algorithm. The accompanied prediction for singular values along the path is done by linearizing \(A(\lambda)=F_ x(x(\lambda),\lambda)\) with respect to \(\lambda\) and considering an appropriate generalized linear eigenvalue problem. It is shown how to exploit the Arnoldi algorithm for solving this eigenvalue problem. The case of turning points is dealt with by taking an augmented problem. The algorithm is applied to a Taylor problem of hydrodynamics: The steady axisymmetric flow of an incompressible viscous fluid between two rotating cylinders. As the parameter \(\lambda\) either the wavelength of the flow or the Reynolds number is considered.
Reviewer: W.Zulehner (Linz)
MSC:
65H20 | Global methods, including homotopy approaches to the numerical solution of nonlinear equations |
65H17 | Numerical solution of nonlinear eigenvalue and eigenvector problems |
65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
35Q30 | Navier-Stokes equations |
76D05 | Navier-Stokes equations for incompressible viscous fluids |