We propose a parallel pseudo-transient continuation algorithm, in conjunction with a Newton-Krylov-Schwarz (NKS) algorithm, for the detection of the critical points of symmetry-breaking bifurcations in sudden expansion flows. One classical approach for examining the stability of a stationary solution to a system of ordinary differential equations (ODEs) is to apply the so-called a method-of-line approach, beginning with some perturbed stationary solution to a system of ODEs and then to investigate its time-dependent response. While the time accuracy is not our concern, the adaptability of time-step size is a key ingredient for the success of the algorithm in accelerating the time-marching process. To allow large time steps, unconditionally stable time integrators, such as the backward Euler's method, are often employed. As a result, the price paid is that at each time step, a large sparse nonlinear system of equations needs to be solved. The NKS is a good candidate solver for a system. Our numerical results obtained from a parallel machine show that our algorithm is robust and efficient and also verify, qualitatively, the bifurcation prediction with published results. Furthermore, imperfect pitchfork bifurcations are observed, especially for the case with a small expansion ratio, in which the occurrence of bifurcation points is delayed due to the stabilization terms in Galerkin/Least squares finite elements on asymmetric, unstructured meshes.
- Domain decomposition
- Incompressible sudden expansion flow
- Parallel computing
- Pseudo-transient continuation