Harnessing time-stepping for cost-effective harmonic resolvent analysis

Harmonic resolvent analysis offers a powerful extension of classical resolvent analysis for modeling flows with dominant periodic motions. The central challenge in computing harmonic resolvent modes is the expanded size of the system, wherein all frequencies of interest are consolidated into a single operator, in contrast to the decoupled system for each frequency in standard resolvent analysis. In this talk, we present an extension of our scalable algorithm, RSVD-$Delta t$, originally designed for computing resolvent modes, to address the computational challenges in harmonic systems while maintaining linear scaling with dimension. Our algorithm combines randomized singular value decomposition with efficient time-stepping techniques to overcome the computational bottleneck within existing methods by computing the action of the harmonic resolvent operator on vectors without explicitly solving a linear system in Fourier space. We validate our algorithm on a toy problem involving a periodic dynamical system around its limit cycle and evaluate its reliability and accuracy by computing the harmonic resolvent modes for a flow passing an airfoil.