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Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow

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Abstract

Stabilization using filters is intended to model and extract the energy lost to resolved scales due to nonlinearity breaking down resolved scales to unresolved scales. This process is highly nonlinear and yet current models for it use linear filters to select the eddies that will be damped. In this report we consider for the first time nonlinear filters which select eddies for damping (simulating breakdown) based on knowledge of how nonlinearity acts in real flow problems. The particular form of the nonlinear filter allows for easy incorporation of more knowledge into the filter process and its computational complexity is comparable to calculating a linear filter of similar form. We then analyze nonlinear filter based stabilization for the Navier–Stokes equations. We give a precise analysis of the numerical diffusion and error in this process.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    William Layton & Catalin Trenchea

  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA

    Leo G. Rebholz

Authors
  1. William Layton
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  2. Leo G. Rebholz
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  3. Catalin Trenchea
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Correspondence to William Layton.

Additional information

Communicated by G.P. Galdi

W. Layton was partially supported by the NSF under grant DMS-0810385; L. G. Rebholz was partially supported by the NSF under grant DMS-0914478; and C. Trenchea was partially supported by the Air Force under grant FA9550-09-1-0058.

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Layton, W., Rebholz, L.G. & Trenchea, C. Modular Nonlinear Filter Stabilization of Methods for Higher Reynolds Numbers Flow. J. Math. Fluid Mech. 14, 325–354 (2012). https://doi.org/10.1007/s00021-011-0072-z

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