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Model reduction of the compressible Euler equations based on properorthogonal decomposition (POD) and Galerkin orthogonality or least-squaresresidual minimization requires the selection of inner product spaces in whichto perform projections and measure norms. The most popular choice is thevector-valued L2({\Omega}) inner product space. This choice, however, yieldsdimensionally-inconsistent reduced-order model (ROM) formulations which oftenlack robustness. In this work, we try to address this weakness by studying aset of dimensionally-consistent inner products with application to thecompressible Euler equations. First, we demonstrate that non-dimensional innerproducts have a positive impact on both POD and Galerkin/least-squares ROMs.Second, we further demonstrate that physics-based inner products based onentropy principles result in drastically more accurate and robust ROMformulations than those based on non-dimensional L2({\Omega}) inner products.Astest cases, we consider the following well-known problems: the one-dimensionalSod shock tube, the two-dimensional Kelvin-Helmholtz instability andtwo-dimensional homogeneous isotropic turbulence.

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