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

Abstract

Summary In this paper, we propose a new evolve‐then‐filter reduced order model (EF‐ROM). This is a regularized ROM (Reg‐ROM), which aims to add numerical stabilization to proper orthogonal decomposition (POD) ROMs for convection‐dominated flows. We also consider the Leray ROM (L‐ROM). These two Reg‐ROMs use explicit ROM spatial filtering to smooth (regularize) various terms in the ROMs. Two spatial filters are used: a POD projection onto a POD subspace (Proj) and a POD differential filter (DF). The four Reg‐ROM/filter combinations are tested in the numerical simulation of the three‐dimensional flow past a circular cylinder at a Reynolds number Re=1000. Overall, the most accurate Reg‐ROM/filter combination is EF‐ROM‐DF. Furthermore, the spatial filter has a higher impact on the Reg‐ROM than the regularization used. Indeed, the DF generally yields better results than Proj for both the EF‐ROM and L‐ROM. Finally, the CPU times of the four Reg‐ROM/filter combinations are orders of magnitude lower than the CPU time of the DNS. Copyright © 2017 John Wiley & Sons, Ltd.

Approximate deconvolution reduced order modeling

“…Later, it was also used in a ROM context to develop regularized ROMs: The ROM-DF was first used in [37] for the 1D Kuramoto-Sivashinsky equation in a periodic setting. The ROM-DF was subsequently used in [38] for the 3D NSE in a general non-periodic setting.…”

“…The goal in the ROM deconvolution problem is to find an approximation of the original flow variable, u, so that the ROM closure problem (30) can be solved. Since G is invertible, at first glance, one just has to use an exact deconvolution (ED), i.e., employ the inverse of the filter G in (38) to solve the deconvolution problem:…”

“…It is well-known in the inverse problem community that the ED is generally a bad idea because the inverse problem (38) is ill-posed: small changes in the data can lead to large changes in the solution [45,50,52]. Indeed, inverting the operator G in (39) results in division by small coefficients of the high-frequency components of the operator G. Thus, any changes in the input (38) translate into large unphysical oscillations in the output (39) [50,52].…”

“…Indeed, inverting the operator G in (39) results in division by small coefficients of the high-frequency components of the operator G. Thus, any changes in the input (38) translate into large unphysical oscillations in the output (39) [50,52]. The ED (39) was carefully investigated in Section 5.2 of [53], where it was shown that, as expected, for input signals (u) of different types, adding noise to the filtered signal (u) and then using the ED (39) results in inaccurate approximations (u E D ), usually in the form of numerical oscillations.…”

“…We emphasize, however, that our approach is fundamentally different, since we explicitly use spatial filters in the actual ROMs, i.e., we modify the mathematical model, not the input data. To our knowledge, explicit ROM spatial filtering has only be used in [14,37,38].…”