This work deals with the complete stability analysis in the case of two dimensional fluid flows searching for steady and Hopf bifurcations.

The stability analysis starts with the computation of steady bifurcations. It is realized by first considering the monitoring of an indicator which is a scalar function. The indicator is computed via a perturbation method: the Asymptotic Numerical Method. Steady bifurcation point corresponds to the zero of this indicator. From this singular point, all the steady bifurcated branches are computed by using the perturbation method.

Then the stability analysis is pursued with the computation of Hopf bifurcations by a hybrid method. This latter consists in coupling a continuation method with a direct Newton solver.

The continuation method allows the bifurcation indicator to be determined using alternating reduced order and full size steps resolution. The quantities coming from the bifurcation indicator are used as initial approximations for the direct method iterations.

Then an augmented system whose solutions are Hopf bifurcation is solved by a direct method of Newton kind.

The examples of the one side and two sides lid driven cavities show the reliability and the efficiency of the proposed numerical tools.

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