2D Burgers equation with large Reynolds number using POD/DEIM and calibration

Abstract

Summary Model order reduction of the two‐dimensional Burgers equation is investigated. The mathematical formulation of POD/discrete empirical interpolation method (DEIM)‐reduced order model (ROM) is derived based on the Galerkin projection and DEIM from the existing high fidelity‐implicit finite‐difference full model. For validation, we numerically compared the POD ROM, POD/DEIM, and the full model in two cases of Re = 100 and Re = 1000, respectively. We found that the POD/DEIM ROM leads to a speed‐up of CPU time by a factor of O(10). The computational stability of POD/DEIM ROM is maintained by means of a careful selection of POD modes and the DEIM interpolation points. The solution of POD/DEIM in the case of Re = 1000 has an accuracy with error O(10−3) versus O(10−4) in the case of Re = 100 when compared with the high fidelity model. For this turbulent flow, a closure model consisting of a Tikhonov regularization is carried out in order to recover the missing information and is developed to account for the small‐scale dissipation effect of the truncated POD modes. It is shown that the computational results of this calibrated ROM exhibit considerable agreement with the high fidelity model, which implies the efficiency of the closure model used. Copyright © 2016 John Wiley & Sons, Ltd.

An evolve‐then‐filter regularized reduced order model for convection‐dominated flows

“…We also extend and investigate the Leray ROM (L-ROM) proposed in [29]. The calibration models in [30][31][32][33][34], on the other hand, utilize the standard ROM equations and only calibrate their coefficients by utilizing a Tikhonov type regularization. Note that these Reg-ROMs are fundamentally different from the calibration approaches used in [30][31][32][33][34].…”

Approximate deconvolution reduced order modeling

“…We believe that this is impressive, given that the new AD-ROM did not use any explicit numerical dissipation mechanisms, whereas the other ROMs did. For high Reynolds number flows, we plan to add numerical dissipation (e.g., time relaxation) to the AD-ROM and perform a thorough comparison with other types of ROMs (e.g., EV-ROMs [3,8,14], Reg-ROMs [37,38] and calibrated ROMs [62][63][64][65]). …”

“…is the vector of coefficients in (64). Comparing approaches (i) and (ii) in terms of computational efficiency, we draw the following conclusions: In approach (i), there is no extra storage required, but the N × N linear system (63) needs to be solved at each time step in the online stage.…”

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

“…We can refer to other works [23][24][25][26] for basic concepts in the POD method, [27][28][29][30][31][32] for using POD technique on numerical solution of PDEs, 33 for application of POD in PDE optimal control, 34-37 for error estimation of the POD method for optimal control problems and to previous studies [38][39][40][41][42][43][44] for other engineering applications. This method generates an optimal set of basis functions (so-called POD basis functions) where each of them has a global support and involves information about the system obtained out of a computational or experimental database (snapshots).…”