The selective frequency damping (SFD) method is an alternative to classical Newtons method to obtain unstable steady-state solutions of dynamical systems. However this method has two main limitations: it does not converge for arbitrary control parame
ters; and when it does converge, the time necessary to reach the steady-state solution may be very long. In this paper we present an adaptive algorithm to address these two issues. We show that by evaluating the dominant eigenvalue of a partially converged steady flow, we can select a control coefficient and a filter width that ensure an optimum convergence of the SFD method. We apply this adaptive method to several classical test cases of computational fluid dynamics and we show that a steady-state solution can be obtained without any a priori knowledge of the flow stability properties.
We present an alternative encapsulated formulation of the Selective Frequency Damping method for finding unstable equilibria of dynamical systems, which is particularly useful when analysing the stability of fluid flows. The formulation makes use of
splitting methods, which means that it can be wrapped around an existing time-stepping code as a black box. The method is first applied to a scalar problem in order to analyse its stability and highlight the roles of the control coefficient $chi$ and the filter width $Delta$ in the convergence (or not) towards the steady-state. Then the steady-state of the incompressible flow past a two-dimensional cylinder at $Re=100$, obtained with a code which implements the spectral/hp element method, is presented.
