Abstract
We discuss the design and implementation of a vector extrapolation method for computing numerical solutions of the steady-state Navier-Stokes equation system. We describe a “proof of concept” implementation of vector extrapolation, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package ( http://www.manchester.ac.uk/ifiss).
Similar content being viewed by others
References
Brezinski, C.: Généralisation de la transformation de Shanks, de la table de Padé et de l’epsilon-algorithm. Calcolo 12, 317–360 (1975)
Brezinski, C., Redivo Zaglia, M.: Extrapolation Methods Theory and Practice. North-Holland, Amsterdam (1991)
Cabay, S., Jackson, L.W.: A polynomial extrapolation method for finding limits and antilimits for vector sequences. SIAM J. Numer. Anal. 13, 734–752 (1976)
Duminil, S., Sadok, H.: Reduced rank extrapolation applied to electronic structure computations. Electron. Trans. Numer. Anal. 38, 347–362 (2011)
Eddy, R.P.: Extrapolation to the limit of a vector sequence. In: Wang, P.C.C. (ed.) Information Linkage Between Applied Mathematics and Industry, pp. 387–396. Academic Press, New-York (1979)
Elman, H., Ramage, A., Silvester, D.: Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow. ACM Trans. Math. Softw. 33, 2–14 (2007)
Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers: Eith Applications in Incompressible Fluid Dynamics. Oxford University Press, Oxford (2005). ISBN: 978-0-19-852868-5; 0-19-852868-X
Gresho, P.M., Gartling, D.K., Torczynski, J.R., Cliffe, K.A., Winters, K.H., Garratt, T.J., Spence, A., Goodrich, J.W.: Is the steady viscous incompressible 2D flow over a backward facing step at Re = 800 stable. Int. J. Numer. Methods Fluids 17, 501–541 (1993)
Jbilou, K.: A general projection algorithm for solving linear systems of equations. Numer. Algoritm. 4, 361–377 (1993)
Jbilou, K., Reichel, L., Sadok, H.: Vector extrapolation enhanced TSVD for linear discrete ill-posed problems. Numer. Algoritm. 51, 195–208 (2009)
Jbilou, K., Sadok, H.: Some results about vector extrapolation methods and related fixed point iterations. J. Comput. Appl. Math. 36, 385–398 (1991)
Jbilou, K., Sadok, H.: Analysis of some vector extrapolation methods for linear systems. Numer. Math. 70, 73–89 (1995)
Jbilou, K., Sadok, H.: LU-implementation of the modified minimal polynomial extrapolation method. IMA J. Numer. Anal. 19, 549–561 (1999)
Jbilou, K., Sadok, H.: Vector extrapolation methods. Applications and numerical comparison. J. Comp. Appl. Math. 122, 149–165 (2000)
Mes̀ina M.: Convergence acceleration for the iterative solution of x = Ax + f. Comput. Methods Appl. Mech. Eng. 10(2), 165–173 (1977)
Pugatchev, B.P.: Acceleration of the convergence of iterative processes and a method for solving systems of nonlinear equations. U.S.S.R. Comput. Math. Math. Phys. 17, 199–207 (1978)
Saad, Y.: Krylov subspace methods for solving large unsymmetric linear systems. Math. Comput. 37, 105–126 (1981)
Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)
Sidi, A.: Extrapolation vs. projection methods for solving linear systems of equations. J. Comput. Appl. Math. 22, 71–88 (1988)
Sidi, A.: Efficient implementation of minimal polynomial and reduced rank extrapolation methods. J. Comput. Appl. Math. 36, 305–337 (1991)
Sidi, A.: Vector extrapolation methods with applications to solution of large systems of equations and to Page rank computations. Comput. Math. Appl. 56, 1–24 (2008)
Sidi, A., Ford, W.F., Smith, D.A.: Acceleration of convergence of vector sequences. SIAM J. Numer. Anal. 23, 178–196 (1986)
Smith, D.A., Ford, W.F., Sidi, A.: Extrapolation methods for vector sequences. SIAM Rev. 29, 199–233 (1987)
Silvester, D., Elman, H., Ramage, A.: Incompressible Flow and Iterative Solver Software (IFISS) version 3.2. Available online at http://www.manchester.ac.uk/ifiss (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Duminil, S., Sadok, H. & Silvester, D. Fast solvers for discretized Navier-Stokes problems using vector extrapolation. Numer Algor 66, 89–104 (2014). https://doi.org/10.1007/s11075-013-9726-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9726-7