On the Symmetrization in POD-Galerkin Model for Linearized Compressible Flows

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 L type to represent kinetic energy. In this paper, our work focuses on the comparison of L inner product and the symmetry inner product introduced by Barone and co-authors (2008). 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 convergency 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 L inner product requires 16 modes for similar convergency. The dynamic behavior described by phase portraits of mode coefficients also gives a cleaner picture when symmetry inner product is used.