We investigate both theoretically and numerically the consistency between the nonlinear discretization in full order models (FOMs) and reduced order models (ROMs) for incompressible flows. To this end, we consider two cases: (i) FOM-ROM consistency, i.e., when we use the same nonlinearity discretization in the FOM and ROM; and (ii) FOM-ROM inconsistency, i.e., when we use different nonlinearity discretizations in the FOM and ROM. Analytically, we prove that while the FOM-ROM consistency yields optimal error bounds, FOM-ROM inconsistency yields additional terms dependent on the FOM divergence error, which prevent the ROM from recovering the FOM as the number of modes increases. Computationally, we consider channel flow around a cylinder and Kelvin-Helmholtz instability, and show that FOM-ROM consistency yields significantly more accurate results than the FOM-ROM inconsistency.

翻译：我们从理论上和数字上调查非线性离散式全序模型(FOMS)和减序型模型(ROMs)在不可压缩流动方面的一致性,为此,我们考虑两种情况:(一) FOM-ROM一致性,即在FOM和ROM中使用同样的非线性离散式;和(二) FOM-ROM不一致,即在FOM和ROM中使用不同的非线性离散型。从分析上看,我们证明FOM-ROM一致性产生最佳误差界限,而FOM-ROM不一致则产生取决于FOM差差错的额外条件,这使得ROM无法随着模式数量的增加而恢复FOM。我们计算,我们考虑在圆筒和Kelvin-Helmholtz周围的通道流动,并表明FOM-ROm一致性产生比FOM-ROm不一致性更准确的结果。