A time-domain preconditioner for the resolvent and harmonic resolvent analyses

Resolvent analysis is a frequency-domain formalism used to study the input-output dynamics of fluid flows in the proximity of time-invariant base flows (e.g., a steady solution of the Navier-Stokes equation, or the temporal mean of a turbulent flow). The harmonic resolvent analysis is analogous, except that the base flow is periodic in time. Both formulations require computing a singular value decomposition of the frequency-domain operator that maps harmonic forcing inputs to post-transient harmonic outputs. In practice, this involves solving large (or extremely large, in the case of the harmonic resolvent) frequency-domain algebraic systems of equations. Furthermore, solving these equations in the frequency domain usually requires computational functionality that is not readily avilable in in-house or open-source time-stepping CFD codes.

We begin to address these issues by observing that the post-transient T-periodic response to a T-periodic forcing input can be computed in the time domain by integrating the Navier-Stokes equations far enough into the future until transients have decayed. (This is a well-known fact.) In order to ``skip the transients'' and make the computation feasible, it is necessary to initialize the time-stepper with an appropriate initial condition q(0) that satisfies q(0) = q(T), where q(T) is the solution at time T. This initial condition can be computed using Newton's method, which we precondition using a novel preconditioner based on a truncated eigendecomposition of the state transition matrix. We use the NekStab package (based on the open-source CFD solver Nek5000) to demonstrate that this approach has the potential to significantly speed up the computations required to perform the resolvent and harmonic resolvent analyses.