Residual-based reduced order models for parameterized Navier–Stokes equations with nonhomogeneous boundary condition

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In this paper, we propose a residual-based reduced-order model (ROM) framework that utilizes available data to increase the ROM accuracy and stability. The available snapshots are utilized to obtain the original ROM systems and their projection coefficients by performing proper orthogonal decomposition. Then a time-parameter varying closure term is added to the original ROM systems to obtain the new ROM systems, and the values of the closure term at discrete time-parameter points with respect to snapshots are evaluated by computing the residual of the original ROM systems with projection coefficients. In an online testing stage, the values of the closure term at unknown time-parameter points are approximated by linear interpolation, and the new ROM systems are utilized to approximate solutions for unseen parameter values. In addition, the closure term in the new ROM systems is instead discarded when we predict systems' evolution outside the time interval with respect to snapshots. Numerical results show that the proposed method not only improves the prediction accuracy of original ROM but also extends the applicability of it.