On the Robustness of Residual Minimization for Constructing POD-Based Reduced-Order CFD Models

Residual minimization is often used in constructing nonlinear reduced-order models (ROMs) for engineering computations. In this paper, it is shown that for unsteady CFD applications, the performance of this method strongly depends on the chosen definition of the residual. To this effect, two common residual definitions are considered first. One is the standard residual associated with the governing high-dimensional discrete equations. The other is obtained by scaling the standard residual with the inverses of the volumes of the cells of the given CFD mesh. This second definition of the residual is common in high-dimensional CFD computations as it maximizes the accuracy of the computed results in the boundary layer regions where the spatial discretization is the finest. Using the proper orthogonal decomposition (POD) method for constructing a reduced-order basis and residual minimization for computing a reduced-order approximation of a CFD solution using this basis, it is shown that both aforementioned residual definitions lead to nonlinear CFD ROMs that may perform poorly. For this reason, a new definition of the residual is proposed for the purpose of nonlinear model reduction. Unlike the two previous definitions, this one is not biased by mesh spacing considerations. More importantly, using Burger’s equation as a model CFD problem with shocks, and the Ahmed body problem as a representative of three-dimensional turbulent flow problems, the proposed definition of the residual is shown to lead to nonlinear ROMs that perform significantly better than their counterparts based of the first two aforementioned definitions of the CFD residual.