On the Robustness of POD-Galerkin ROMs with Symmetrizable Governing Equations

Evolution of Reduced-Order Models (ROMs) to successfully respond to a variety of flow conditions is crucial to their adjustment to flow control demands. This study is focused on the influence of inner product definition on the robustness of POD-Galerkin models. The quality of Symmetry inner product in improving stability of compressible flow ROMs has been demonstrated in previous studies. The current work shows that, when unsteady discontinuities in supersonic flows resolved by original high-fidelity simulation are not resolved in a lower-order space over a limited number of leading POD modes through Galerkin ROMs, using the Symmetry inner product notably restricts the amount by which the ROM deviates from its ideal stable and accurate state. The L2 ROM on the other hand, easily leaps toward extremely unstable conditions in a low-order space. Meanwhile, when the originally unstable linear and nonlinear ROMs based on L2 and Symmetry inner products are stabilized by an Eigenvalue Reassignment method empowered by Particle Swarm Intelligence, the Symmetry ROMs have shown to be more robust against suboptimal control laws.