On the Stability of Reduced-Order Linearized Computational Fluid Dynamics Models Based on POD and Galerkin Projection: Descriptor vs Non-Descriptor Forms

Abstract

The Galerkin projection method based on modes generated by the Proper Orthogonal Decomposition (POD) technique is very popular for the dimensional reduction of linearized Computational Fluid Dynamics (CFD) models, among many other typically high-dimensional models in computational engineering. This, despite the fact that it cannot guarantee neither the optimality nor the stability of the Reduced- Order Models (ROMs) it constructs. Short of proposing any variant of this model order reduction method that guarantees the stability of its outcome, this paper contributes a best practice to its application to the construction of linearized CFD ROMs. It begins by establishing that whereas the solution snapshots computed using the descriptor and non-descriptor forms of the discretized Euler or Navier-Stokes equations are identical, the ROMs obtained by reducing these two alternative forms of the governing equations of interest are different. Focusing next on compressible fluid-structure interaction problems associated with computational aeroelasticity, this paper shows numerically that the POD-based fluid ROMs originating from the non-descriptor form of the governing linearized CFD equations tend to be unstable, but their counterparts originating from the descriptor form of these equations are typically stable and reliable for aeroelastic applications. Therefore, this paper argues that whereas many computations are performed in CFD codes using the non-descriptor form of discretized Euler and/or Navier-Stokes equations, POD-based model reduction in these codes should be performed using the descriptor form of these equations.