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Several nonlinear model reduction techniques are compared for the three casesof the non-parallel version of the Kuramoto-Sivashinsky equation, the transientregime of flow past a cylinder at $Re=100$ and fully developed flow past acylinder at the same Reynolds number. The linear terms of the governingequations are reduced by Galerkin projection onto a POD basis of the flowstate, while the reduced nonlinear convection terms are obtained either by aGalerkin projection onto the same state basis, by a Galerkin projection onto aPOD basis representing the nonlinearities or by applying the Discrete EmpiricalInterpolation Method (DEIM) to a POD basis of the nonlinearities. The qualityof the reduced order models is assessed as to their stability, accuracy androbustness, and appropriate quantitative measures are introduced and compared.In particular, the properties of the reduced linear terms are compared to thoseof the full-scale terms, and the structure of the nonlinear quadratic terms isanalyzed as to the conservation of kinetic energy. It is shown that all threereduction techniques provide excellent and similar results for the cases of theKuramoto-Sivashinsky equation and the limit-cycle cylinder flow. For the caseof the transient regime of flow past a cylinder, only the pure Galerkintechniques are successful, while the DEIM technique produces reduced-ordermodels that diverge in finite time.

Nonlinear model reduction: a comparison between POD-Galerkin and POD-DEIM methods