Projection methods for reduced order models of compressible flows

Two different projection methods, Galerkin projection and direct projection, are developed for reduced-order modeling applications. The projection methods are used to identify low-dimensional systems of ordinary differential equations to represent the dynamics of a compressible, two-dimensional, inviscid flow-field under oscillatory forcing. Proper orthogonal decomposition is used to identify a small number of fluid modes to serve as the basis functions for the projections. Performance is evaluated relative to a high-order numerical model in terms of accuracy, order reduction, and computational efficiency. The treatment of boundary conditions, and stability of the reduced-order model are addressed in detail. The methods developed in this paper are suitable for general application to the Euler equations. With the addition of dissipation parameters, both the Galerkin projection and direct projection methods are tractable, stable, and properly treat the boundary conditions.