Commutation error in reduced order modeling of fluid flows
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Commutation error in reduced order modeling of fluid flows Birgul Koc1 · Muhammad Mohebujjaman2 · Changhong Mou1 · Traian Iliescu1 Received: 1 October 2018 / Accepted: 3 October 2019 / © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract For reduced order models (ROMs) of fluid flows, we investigate theoretically and computationally whether differentiation and ROM spatial filtering commute, i.e., whether the commutation error (CE) is nonzero. We study the CE for the Laplacian and two ROM filters: the ROM projection and the ROM differential filter. Furthermore, when the CE is nonzero, we investigate whether it has any significant effect on ROMs that are constructed by using spatial filtering. As numerical tests, we use the Burgers equation with viscosities ν = 10−1 and ν = 10−3 and a 2D flow past a circular cylinder at Reynolds numbers Re = 100 and Re = 500. Our investigation (i) measures the size of the CE in these test problems and (ii) shows that the CE has a significant effect on ROM development for high viscosities, but not so much for low viscosities. Keywords Reduced order model · Spatial filter · Commutation error · Data-driven model Mathematics Subject Classification (2010) 65M60 · 76F65
1 Introduction 1.1 Motivation and prior work Reduced order models (ROMs) [11, 15, 34] have been used for decades in the numerical simulation of fluid flows [2, 4, 14, 16, 27, 40, 41]. If the ROM dimension is
Communicated by: Anthony Nouy Changhong Mou and Traian Iliescu are partially supported by DMS-1821145 Birgul Koc
[email protected]
Extended author information available on the last page of the article.
B. Koc et al.
high enough to capture the relevant flow features, ROMs yield efficient and relatively accurate approximations of the underlying flows. However, to ensure a low computational cost, the ROM dimension is often lower than the dimension required by an accurate numerical simulation. This happens, for example, in turbulent flows, where the ROM dimension is generally significantly lower than the dimension needed to capture all the relevant flow features [8, 9, 12, 16, 30, 39]. In these challenging simulations, the low dimensional ROMs often yield inaccurate results. The general explanation for these inaccurate results is that the ROMs fail to account for the interaction between the resolved ROM modes and the unresolved ROM modes that are discarded in the often drastic ROM truncation [9, 13, 28, 29, 42, 44, 45]. Thus, when the ROM dimension is too low to capture the relevant flow features, ROMs are generally supplemented with a Correction term [1, 3, 12, 13, 16, 21, 27, 30, 37, 42]. In our recent work [44], we have shown that this Correction term can be explicitly calculated and modeled with the available data by using the ROM projection as a ROM spatial filter. (In Section 3.1, we present two examples of ROM spatial filters: the ROM differential filter and the ROM projection filter.) We note that ROM spatial filtering has also been used to develop large eddy simulation ROMs, e.g., approx
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