Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 2, pp. 317-348.

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.

DOI: 10.1051/m2an/2012029
Classification: 35K20,  35K55,  65M15,  65M60
Keywords: reduced-basis methods, parametrized pdes, nonlinear PDEs, Burgers equation
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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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {317--348},
publisher = {EDP-Sciences},
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year = {2013},
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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 - Modélisation Mathématique et Analyse Numérique, Volume 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] I. Babuska, The finite element method with penalty. Math. Comput. 27 (1973) 221-228. | MR | Zbl

[3] J.W. Barrett and C.M. Elliott, Finite element approximation of the Dirichlet problem using the boundary penalty method. Numer. Math. 49 (1986) 343-366. | MR | Zbl

[4] A. Buffa, Y. Maday, A.T. Patera, C. Prud'Homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis. ESAIM : M2AN (2009). | Numdam | Zbl

[5] A. Chatterjee, An introduction to the proper orthogonal decomposition. Current Sci. 78 (2000) 808-817.

[6] V. Girault and P.A. Raviart, 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] M.A. Grepl, Reduced-Basis Approximation and A Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. thesis, Massachusetts Institute of Technology (2005).

[10] M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. ESAIM : M2AN 39 (2005) 157-181. | Numdam | MR | Zbl

[11] M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM : M2AN 41 (2007) 575-605. | Numdam | MR | Zbl

[12] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM : M2AN 42 (2008) 277-302. | Numdam | MR

[13] J.C. Helton, J.D. Johnson, C.J. Sallaberry and C.B. Storlie, Survey of sampling-based methods for uncertainty and sensitivity analysis. Reliability Engineering and System Safety 91 (2006) 1175-1209.

[14] E. Hopf, The partial differential equation ut + uux = μxx.Commun. Pure Appl. Math. 3 (1950) 201-230. | MR | Zbl

[15] D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, 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] N. Jung, B. Haasdonk and D. Kroner, Reduced Basis Method for quadratically nonlinear transport equations. Int. J. Comput. Sci. Math. 2 (2009) 334-353. | MR | Zbl

[17] D.J. Knezevic and A.T. Patera, 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] N.C. Nguyen, K. Veroy and A.T. Patera, Certified real-time solution of parametrized partial differential equations. Handbook Mater. Mod. (2005) 1523-1558.

[19] N.C. Nguyen, G. Rozza and A.T. Patera, Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers's equation. Calcolo 46 (2009) 157-185. | MR | Zbl

[20] J. Nocedal and S.J. Wright, Numerical optimization. Springer-Verlag (1999). | MR | Zbl

[21] A.M. Quarteroni and A. Valli, Numerical approximation of partial differential equations. Springer (2008). | Zbl

[22] D.V. Rovas, L. Machiels and Y. Maday, Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal. 26 (2006) 423. | MR | Zbl

[23] A. Saltelli, K. Chan and E.M. Scott, Sensitivity analysis. Wiley, New York (2000). | MR | Zbl

[24] J.C. Strikwerda, Finite difference schemes and partial differential equations. Society for Industrial Mathematics (2004). | MR | Zbl

[25] K. Veroy and A.T. Patera, 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] K. Veroy, C. Prud'Homme and A.T. Patera, 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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