Efficient Adjoint Petrov-Galerkin reduced order models for fluid flows governed by the semi-discrete incompressible Navier-Stokes equations

This research paper investigates the Adjoint Petrov-Galerkin (APG) method for reduced order models (ROMs) and fluid dynamics governed by the semi-discrete incompressible Navier-Stokes equations. The Adjoint Petrov-Galerkin ROM, derived using the Mori-Zwanzig formalism, demonstrates superior accuracy and stability compared to standard Galerkin ROMs. However, challenges arise due to the time invariance of the test basis vectors, resulting in high computational requirements. To address this, we introduce a new efficient Adjoint Petrov-Galerkin (eAPG) ROM formulation, extending its application to the semi-discrete incompressible Navier-Stokes equations. While offline-online decomposition is well established in reduced order modeling, the novelty here lies in its adaptation to the APG framework: by exploiting the polynomial structure of the governing equations, the eAPG formulation eliminates the need for repeated test basis evaluations in the online stage. This reorganization yields an exact offline-online split that improves computational efficiency in comparison to the general APG-ROM formulation. A novel approach to augmenting the memory length, a critical factor influencing the stability and accuracy of the APG-ROM, is introduced, employing a data-driven optimization. Numerical results for the 3D turbulent flow around a circular cylinder demonstrate the efficacy of the proposed approach. Error measures and computational cost evaluations, considering metrics such as floating point operations, memory usage, and simulation time, provide a comprehensive analysis.