Abstract
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.