Space-time model reduction using SPOD modes

We present a space-time reduced-order model (ROM) based on spectral proper orthogonal decomposition (SPOD) modes and apply it to the (nonlinear) Navier-Stokes equations (NSE). The NSE, linearized around a mean flow, can provide a surprising amount of insight into the behavior of the fully nonlinear flow. The left singular vectors of the resolvent operator at a given frequency, e.g., often give accurate predictions for the most prevalent flow structures at that frequency, i.e., the SPOD modes. Linear time-invariant systems are well-understood analytically, and in previous work, we leveraged this to derive a space-time ROM that returns the SPOD coefficients corresponding to a time interval [0,T] in terms of the initial condition and forcing to the linear system. These coefficients provide an effective reduced representation of the solution, and the method yielded ~100-fold lower error than benchmarks like POD-Galerkin projection at the same cost. In this work, we account for nonlinearity and apply the method to the incompressible NSE. Nonlinearity is handled by first computing the SPOD coefficients for the linearized system, then calculating the nonlinearity that would result from the corresponding trajectory, applying this as a forcing to the system, and iterating until convergence. In the tests carried out so far on simple PDEs, convergence is achieved after just a few iterations. We implement the method in the spectral element code Nek5000 and test its accuracy relative to POD-Galerkin projection.