On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vector-valued L2(Ω) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this work, we try to address this weakness by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. First, we demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. Second, we further demonstrate that physics-based inner products based on entropy principles result in drastically more accurate and robust ROM formulations than those based on non-dimensional L2(Ω) inner products. As test cases, we consider the following problems: the one-dimensional Sod shock tube, a two-dimensional Riemann problem, a two-dimensional Kelvin-Helmholtz instability, and two-dimensional homogeneous isotropic turbulence.