| Summary: | 碩士 === 國立中央大學 === 數學研究所 === 98 === In this study, we focus in investigating the relation between the (linear) stability of stationary solutions and pitchfork bifurcations of incompressible flows, and detect the critical points of symmetry-breaking phenomena. First, a stabilized finite element method is used to discretize the 2D Navier-Stokes equations on the spatial domain for the unsteady, viscous, incompressible flow problem. There are two approaches used to determine the behavior of the solution. One is via numerical time integration. Another is to locate the steady-state solutions and then to make the linear stability analysis by computing eigenvalues of a corresponding generalized eigenvalue problem, for which an implicit Arnoldi method with the Cayley transformation is used. In addition, it is also an important issue that how to choose the parameters of the Cayley transformation such that the convergence of the linear system would be better. Finally, we show a parallel performance of SuperLU, a great parallelable algorithm which is used to solve the linear system.
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