Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows

The proper orthogonal decomposition (POD) method as a systematic technique to analyze parameter-dependent problems may be inappropriate as a practical tool for generating low-dimensional models. We propose a weighted POD (w-POD) as an alternative to give higher priority to low-energetic or important modes by simply weighting. A predefined POD (p-POD) is suggested, where modes are selected not only on the basis of energy but also on some a priori knowledge of the system. The techniques are tested on a flow problem undergoing steady or unsteady transition but may in general be applied to nonlinear systems of discretized PDEs closed by additional linear algebraic equations. Various residual functions to measure the quality of the reduced models are introduced. They provide lower limits of residual energy to ensure an adequate physical representation for a given modeling. It is shown that modes extracted locally may generally not contain information regarding the global dynamics, and the dynamics may not be correctly predicted over a longer range of model parameters. It is demonstrated that by weighting and predefining base vectors it is possible to improve the POD technique's capability to generate low-dimensional models.