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Commutation error in reduced order modeling of fluid flows

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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.

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Acknowledgements

We thank the three anonymous reviewers for their constructive comments and suggestions, which significantly improved the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24061, USA

    Birgul Koc, Changhong Mou & Traian Iliescu

  2. Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Muhammad Mohebujjaman

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  1. Birgul Koc
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Correspondence to Birgul Koc.

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Communicated by: Anthony Nouy

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Changhong Mou and Traian Iliescu are partially supported by DMS-1821145

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Koc, B., Mohebujjaman, M., Mou, C. et al. Commutation error in reduced order modeling of fluid flows. Adv Comput Math 45, 2587–2621 (2019). https://doi.org/10.1007/s10444-019-09739-0

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