We present a reduced basis offline/online procedure for viscous Burgers initial boundary value problem, enabling efficient approximate computation of the solutions of this equation for parametrized viscosity and initial and boundary value data. This procedure comes with a fast-evaluated rigorous error bound certifying the approximation procedure. Our numerical experiments show significant computational savings, as well as efficiency of the error bound.
Mots clés : reduced-basis methods, parametrized pdes, nonlinear PDEs, Burgers equation
@article{M2AN_2013__47_2_317_0,
author = {Janon, Alexandre and Nodet, Ma\"elle and Prieur, Cl\'ementine},
title = {Certified reduced-basis solutions of viscous {Burgers} equation parametrized by initial and boundary values},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {317--348},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
doi = {10.1051/m2an/2012029},
mrnumber = {3021689},
zbl = {1272.35016},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012029/}
}
TY - JOUR AU - Janon, Alexandre AU - Nodet, Maëlle AU - Prieur, Clémentine TI - Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 317 EP - 348 VL - 47 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012029/ DO - 10.1051/m2an/2012029 LA - en ID - M2AN_2013__47_2_317_0 ER -
%0 Journal Article %A Janon, Alexandre %A Nodet, Maëlle %A Prieur, Clémentine %T Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 317-348 %V 47 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012029/ %R 10.1051/m2an/2012029 %G en %F M2AN_2013__47_2_317_0
Janon, Alexandre; Nodet, Maëlle; Prieur, Clémentine. Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 2, pp. 317-348. doi : 10.1051/m2an/2012029. http://www.numdam.org/articles/10.1051/m2an/2012029/
[1] ARPACK : Arnoldi Package, available on http://www.caam.rice.edu/software/ARPACK/.
[2] , The finite element method with penalty. Math. Comput. 27 (1973) 221-228. | MR | Zbl
[3] and , Finite element approximation of the Dirichlet problem using the boundary penalty method. Numer. Math. 49 (1986) 343-366. | MR | Zbl
[4] , , , and , A priori convergence of the greedy algorithm for the parametrized reduced basis. ESAIM : M2AN (2009). | Numdam | Zbl
[5] , An introduction to the proper orthogonal decomposition. Current Sci. 78 (2000) 808-817.
[6] and , Finite element methods for Navier-Stokes equations, Springer Series in Comput. Math. 5 (1986). | MR | Zbl
[7] GLPK : GNU Linear Programming Kit, available on http://www.gnu.org/software/glpk/.
[8] GOMP : An OpenMP implementation for GCC, available on http://gcc.gnu.org/projects/gomp/.
[9] , Reduced-Basis Approximation and A Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology (2005).
[10] and , A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM : M2AN 39 (2005) 157-181. | Numdam | MR | Zbl
[11] , , and , Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM : M2AN 41 (2007) 575-605. | Numdam | MR | Zbl
[12] and , Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM : M2AN 42 (2008) 277-302. | Numdam | MR
[13] , , and , Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering and System Safety 91 (2006) 1175-1209.
[14] , The partial differential equation ut + uux = μxx.Commun. Pure Appl. Math. 3 (1950) 201-230. | MR | Zbl
[15] , , and , A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math. 345 (2007) 473-478. | MR | Zbl
[16] , and , Reduced Basis Method for quadratically nonlinear transport equations. Int. J. Comput. Sci. Math. 2 (2009) 334-353. | MR | Zbl
[17] and , A certified reduced basis method for the Fokker-Planck equation of dilute polymeric fluids : FENE dumbbells in extensional flow. SIAM J. Sci. Comput. 32 (2010) 793-817. | MR | Zbl
[18] , and , Certified real-time solution of parametrized partial differential equations. Handbook Mater. Mod. (2005) 1523-1558.
[19] , and , Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers's equation. Calcolo 46 (2009) 157-185. | MR | Zbl
[20] and , Numerical optimization. Springer-Verlag (1999). | MR | Zbl
[21] and , Numerical approximation of partial differential equations. Springer (2008). | Zbl
[22] , and , Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423. | MR | Zbl
[23] , and , Sensitivity analysis. Wiley, New York (2000). | MR | Zbl
[24] , Finite difference schemes and partial differential equations. Society for Industrial Mathematics (2004). | MR | Zbl
[25] and , Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations : Rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47 (2005) 773-788. | MR | Zbl
[26] , and , Reduced-basis approximation of the viscous Burgers equation : rigorous a posteriori error bounds. C. R. Math. 337 (2003) 619-624. | MR | Zbl
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