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.

我们从理论上和数值上研究了全阶模型 (FOM) 和降阶模型中的非线性离散化之间的一致性（ROM）用于不可压缩的流动。为此，我们考虑两种情况： (i) FOM-ROM 一致性，即当我们在 FOM 和 ROM 中使用相同的非线性离散化时；(ii) FOM-ROM 不一致，即当我们在 FOM 和 ROM 中使用不同的非线性离散化时。通过分析，我们证明了虽然 FOM-ROM 一致性产生了最佳误差界限，但 FOM-ROM 不一致性产生了依赖于 FOM 发散误差的附加项，这会阻止 ROM 随着模式数量的增加而恢复 FOM。在计算上，我们考虑了圆柱体周围的通道流动和开尔文-亥姆霍兹不稳定性，并表明 FOM-ROM 一致性比 FOM-ROM 不一致性产生了更准确的结果。