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

中文翻译：

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非线性模型缩减：POD-Galerkin 和 POD-DEIM 方法之间的比较

针对 Kuramoto-Sivashinsky 方程的非平行版本、在 $Re=100$ 处流过圆柱体的瞬态状态和在相同雷诺数下流过圆柱体的完全发展流的三种情况，比较了几种非线性模型缩减技术. 控制方程的线性项通过在流动状态的 POD 基上的 Galerkin 投影来减少，而减少的非线性对流项通过在相同状态基上的 Galerkin 投影，通过在 POD 基上的 Galerkin 投影来表示非线性或通过将离散经验插值方法 (DEIM) 应用于非线性的 POD 基础。对降阶模型的质量进行稳定性、准确性和鲁棒性的评估，并引入和比较适当的定量措施。特别是将简化的线性项的性质与满量程项的性质进行了比较，并分析了非线性二次项的动能守恒结构。结果表明，对于 Kuramoto-Sivashinsky 方程和极限循环气缸流的情况，所有三种减少技术都提供了出色且相似的结果。对于流过圆柱体的瞬态状态的情况，只有纯伽辽金技术是成功的，而 DEIM 技术会产生在有限时间内发散的降阶模型。结果表明，对于 Kuramoto-Sivashinsky 方程和极限循环气缸流的情况，所有三种减少技术都提供了出色且相似的结果。对于流过圆柱体的瞬态状态的情况，只有纯伽辽金技术是成功的，而 DEIM 技术产生在有限时间内发散的降阶模型。结果表明，对于 Kuramoto-Sivashinsky 方程和极限循环气缸流的情况，所有三种减少技术都提供了出色且相似的结果。对于流过圆柱体的瞬态状态的情况，只有纯伽辽金技术是成功的，而 DEIM 技术产生在有限时间内发散的降阶模型。