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 FOM-ROM inconsistency.

翻译：我们从理论上和数字上调查了非线性离散式全序模型(FOMs)和减少订单模型(ROMs)在不可压缩流动方面的一致性,为此,我们考虑了两个案例:(一) FOM-ROM一致性,即在FOM和ROM中使用同样的非线性离散式数据时;和(二) FOM-ROM不一致性数据,即在FOM和ROM中使用不同的非线性离散式数据。分析上,我们证明FOM-ROM一致性产生最佳误差界限时,FOM-ROM不一致产生更多取决于FOM差差错的条件,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周围的通道流动情况,并表明FOM-ROM一致性产生比FOM-ROM不一致性数据更准确得多的结果。