Nonlinear and nonaffine terms in parametric partial differential equations can potentially lead to a computational cost of a reduced order model (ROM) that is comparable to the cost of the original full order model (FOM). To address this, the Reduced Residual Reduced Over-Collocation method (R2-ROC) is developed as a hyper-reduction method within the framework of the reduced basis method in the collocation setting. R2-ROC greedily selects two sets of reduced collocation points based on the (generalized) empirical interpolation method for both solution snapshots and residuals, thereby avoiding the computational inefficiency. The vanilla R2-ROC method can face instability when applied to parametric fluid dynamic problems. To address this, an adaptive enrichment strategy has been proposed to stabilize the ROC method. However, this strategy can involve in an excessive number of reduced collocation points, thereby negatively impacting online efficiency. To ensure both efficiency and accuracy, we propose an adaptive time partitioning and adaptive enrichment strategy-based ROC method (AAROC). The adaptive time partitioning dynamically captures the low-rank structure, necessitating fewer reduced collocation points being sampled in each time segment. Numerical experiments on the parametric viscous Burgers' equation and lid-driven cavity problems demonstrate the efficiency, enhanced stability, and accuracy of the proposed AAROC method.

翻译：参数化偏微分方程中的非线性与非仿射项可能导致降阶模型的计算成本与原全阶模型相当。为解决此问题，在配点框架下的降阶基方法中，我们发展了降阶残差降阶过配点法作为一种超降阶方法。R2-ROC 基于（广义）经验插值法，针对解快照与残差贪婪地选取两组降阶配点，从而规避计算低效性。原始 R2-ROC 方法在应用于参数化流体动力学问题时可能面临不稳定性。为此，已有研究提出自适应增强策略以稳定 ROC 方法。然而，该策略可能导致降阶配点数量过多，进而影响在线计算效率。为兼顾效率与精度，我们提出一种基于自适应时间划分与自适应增强策略的 ROC 方法。自适应时间划分能动态捕捉低秩结构，使得每个时间段内所需采样的降阶配点数量减少。在参数化粘性 Burgers 方程与顶盖驱动方腔流问题上的数值实验验证了所提 AAROC 方法的高效性、增强的稳定性与计算精度。