TY - JOUR

T1 - On a certified smagorinsky reduced basis turbulence model

AU - Ballarin, Francesco

AU - Rebollo, Tomás Chacón

AU - Ávila, Enrique Delgado

AU - Marmol, Macarena Gómez

AU - Rozza, Gianluigi

PY - 2017

Y1 - 2017

N2 - In this work we present a reduced basis Smagorinsky turbulence model for steady ows. We approximate the nonlinear eddy diffusion term using the empirical interpolation method (cf. [M. A. Grepl et al., ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 575-605; Barrault et al., C. R. Acad. Sci. Paris Sffer. I Math., 339 (2004), pp. 667-672]) and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for a Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on [S. Deparis, SIAM J. Sci. Comput., 46 (2008), pp. 2039-2067] and [A. Manzoni, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1199-1226], according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the nonlinear eddy diffusion term. We present some numerical tests, programmed in FreeFem++ (cf. [F. Hecht, J. Numer. Math., 20 (2012), pp. 251-265]), in which we show a speedup on the computation by factor larger than 1000 in benchmark two-dimensional ows.

AB - In this work we present a reduced basis Smagorinsky turbulence model for steady ows. We approximate the nonlinear eddy diffusion term using the empirical interpolation method (cf. [M. A. Grepl et al., ESAIM Math. Model. Numer. Anal., 41 (2007), pp. 575-605; Barrault et al., C. R. Acad. Sci. Paris Sffer. I Math., 339 (2004), pp. 667-672]) and the velocity-pressure unknowns by an independent reduced-basis procedure. This model is based upon an a posteriori error estimation for a Smagorinsky turbulence model. The theoretical development of the a posteriori error estimation is based on [S. Deparis, SIAM J. Sci. Comput., 46 (2008), pp. 2039-2067] and [A. Manzoni, ESAIM Math. Model. Numer. Anal., 48 (2014), pp. 1199-1226], according to the Brezzi-Rappaz-Raviart stability theory, and adapted for the nonlinear eddy diffusion term. We present some numerical tests, programmed in FreeFem++ (cf. [F. Hecht, J. Numer. Math., 20 (2012), pp. 251-265]), in which we show a speedup on the computation by factor larger than 1000 in benchmark two-dimensional ows.

KW - A posteriori error estimation

KW - Empirical interpolation method

KW - Reduced basis method

KW - Steady Smagorinsky model

KW - A posteriori error estimation

KW - Empirical interpolation method

KW - Reduced basis method

KW - Steady Smagorinsky model

UR - http://hdl.handle.net/10807/174167

U2 - 10.1137/17M1118233

DO - 10.1137/17M1118233

M3 - Article

VL - 55

SP - 3047

EP - 3067

JO - SIAM Journal on Numerical Analysis

JF - SIAM Journal on Numerical Analysis

SN - 0036-1429

ER -