In this paper, we propose a new stabilized projection-based POD-ROM for the numerical simulation of incompressible flows. The new method draws inspiration from successful numerical stabilization techniques used in the context of Finite Element (FE) methods, such as Local Projection Stabilization (LPS). In particular, the new LPS-ROM is a velocity-pressure ROM that uses pressure modes as well to compute the reduced order pressure, needed for instance in the computation of relevant quantities, such as drag and lift forces on bodies in the flow. The new LPS-ROM circumvents the standard discrete inf-sup condition for the POD velocity-pressure spaces, whose fulfillment can be rather expensive in realistic applications in Computational Fluid Dynamics (CFD). Also, the velocity modes does not have to be neither strongly nor weakly divergence-free, which allows to use snapshots generated for instance with penalty or projection-based stabilized methods. The numerical analysis of the fully Navier-Stokes discretization for the new LPS-ROM is presented, by mainly deriving the corresponding error estimates. Numerical studies are performed to discuss the accuracy and performance of the new LPS-ROM on a two-dimensional laminar unsteady flow past a circular obstacle.

中文翻译：

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基于投影的不可压缩流稳定 POD-ROM 的数值分析

在本文中，我们提出了一种新的基于稳定投影的 POD-ROM，用于不可压缩流动的数值模拟。新方法的灵感来自于有限元 (FE) 方法中使用的成功数值稳定技术，例如局部投影稳定 (LPS)。特别是，新的 LPS-ROM 是一种速度-压力 ROM，它也使用压力模式来计算降阶压力，例如在计算相关量时需要，例如流中物体的阻力和升力。新的 LPS-ROM 绕过了 POD 速度-压力空间的标准离散 inf-sup 条件，在计算流体动力学 (CFD) 的实际应用中，该条件的实现可能相当昂贵。此外，速度模式不必既无强发散也无弱发散，这允许使用例如通过惩罚或基于投影的稳定方法生成的快照。对新 LPS-ROM 的完全 Navier-Stokes 离散化进行了数值分析，主要是通过推导相应的误差估计。进行数值研究以讨论新型 LPS-ROM 在经过圆形障碍物的二维层流不稳定流上的准确性和性能。