| Reduced-order models (ROM) based on POD-Galerkin projection have shown success in many problems since the approach was introduced approximately two decades ago. Traditionally, the inner product used in computation is L2 type to represent kinetic energy. In this research, our work focuses on the comparison of L2 inner product and the symmetry inner product pioneered by Barone and co-authers (2008) for the stability of ROMs for linear acoustic problems. On the first part, the numerical simulation and analysis are based on a linear acoustic problem controlled by the linearized Euler equation, which, without viscosity, is more sensitive to instability than the Navier-Stokes equation. In our study, besides the stability advantage noticed by Barone's group, much better accuracy and convergence are also shown in the ROM using symmetry inner product. In the test case, symmetry inner product allows to use only 8 modes for the model results to match the exact solution, while L2 inner product requires 16 modes for similar convergence. The dynamic behavior described by phase portraits of mode coefficients also gives a cleaner picture when symmetry inner product is used.;After observing such an improvement in stability, accuracy, and convergence of the linear ROMs using symmetry inner products, we wanted to take advantage of the benefits for non-linear equations. The regular symmetry inner product model is not applicable for non-linear equations and we have applied special treatments to make it possible. We have tested two cases, 1D shock tube and 2D ideal blast wave, and for both cases symmetry inner product shows much better accuracy and convergence compare to the L2 inner product ROMs. The basic idea is to apply the above linear approach using symmetry inner product directly on nonlinear equations, so that the "symmetrized" nonlinear ROM has a symmetrized and stabilized linear term and other nonlinear terms derived from the same symmetry inner product. The obvious benefit is to have a stable linear term in the nonlinear ROM, which has stability often dominated by linear terms. In fact, our study has found that better accuracy is also shown in symmetrized nonlinear ROMs. In addition, the reconstructed flow comparisons show that the new approach's ROM is closer to the DNS data. |
