Computation of Hopf bifurcations coupling reduced order models and the asymptotic numerical method

Abstract

This work deals with the computation of Hopf bifurcation points in the framework of two-dimensional fluid flows. These bifurcation points are determined by using a Hybrid method [1] which associates an indicator curve and a Newton method. The indicator provides initial values for the Newton method. As the calculus of this indicator is time consuming, we suggest using an algorithm to save computational time. This algorithm alternates reduced order and full size step resolution which are all carried out by using a pertubation method. Hence, the computed vectors on the full size problem are used to define the reduced order model. As the low-dimensional model has a finite validity range, we propose a simple criterion which makes it possible to know when the basis has to be updated. The latter phase is carried out by going through a new full step which permits to build a new basis and, thus, compute a supplementary part of the indicator curve. Some numerical tests, such as the classical lid-driven cavity or the flow in a channel, permit to fix the optimum values of the parameters for the proposed method. The objective of this study is to save computational time without modifying the performance of the Hybrid method initially introduced in Ref. [1]. These numerical methods are applied to 2D fluid flows (flow in a channel and the 2D lid-driven cavity). Our conclusions, therefore, hold only for these kinds of problem.