On the importance of numerical error in constructing pod-based reduced-order models of nonlinear fluid flows
The proper orthogonal decomposition (POD) provides a useful method for studying nonlinear fluid dynamics; however, the construction and use of the POD basis modes for Reduced Order Modeling (ROM) introduces several sources of error which can jeopardize the fidelity of the resulting ROM simulations. Given the chaotic nature of turbulent fluid flows, an understanding of these sources of error and their influence on the simulated dynamics is important to the successful implementation of the POD method. In particular, application of the divergence theorem to the pressure term in the Galerkin formulation of the Navier-Stokes equations presents a clear requirement for when the pressure term can be neglected in constructing the ROM. In the present paper, sources of error are identified which call into question if a POD basis for an incompressible flow is divergence-free and if the pressure term can thus be correctly neglected. Methods for mitigating or avoiding these sources of error will be explored.