A Robust Preconditioned Iterative Method for the Navier-Stokes Equations with High Reynolds Numbers
Applied and Computational Mathematics
Volume 6, Issue 4, August 2017, Pages: 202-207
Received: Aug. 13, 2017;
Published: Aug. 14, 2017
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Josaphat Uvah, Department of Mathematics and Statistics, University of West Florida, Pensacola, USA
Jia Liu, Department of Mathematics and Statistics, University of West Florida, Pensacola, USA
Lina Wu, Department of Mathematics, Borough of Manhattan Community College, The City University of New York, New York, USA
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In this paper, we proposed a new solver for the Navier-Stokes equations coming from the channel flow with high Reynolds number. We use the preconditioned Krylov subspace iterative methods such as Generalized Minimum Residual Methods (GMRES). We consider the variation of the Hermitian and Skew-Hermitian splitting to construct the preconditioner. Convergence of the preconditioned iteration is analyzed. We can show that the proposed preconditioner has a robust behavior for the Navier-Stokes problems in variety of models. Numerical experiments show the robustness and efficiency of the preconditioned GMRES for the Navier-Stokes problems with Reynolds numbers up to ten thousands.
Preconditioning, GMRES, Navier-Stokes, High Reynolds Number, Iterative Methods
To cite this article
A Robust Preconditioned Iterative Method for the Navier-Stokes Equations with High Reynolds Numbers, Applied and Computational Mathematics.
Vol. 6, No. 4,
2017, pp. 202-207.
D. Acheson Elementary Fluid Dynamics. Oxford University Press, Oxford, 1990.
G. Batchelor, An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 2000.
M. Benzi and J. Liu, An Efficient Solver for the Incompressible Navier-Stokes Equations in Rotation Form, SIAM J. Scientific Computing, 29 (2007), pp. 1959-1981.
H. Elman, D. Silvester, and A. Wathen, Finite Elements and Fast Iterative Solvers with applications in incompressible fluid dynamics. Oxford University Press, Oxford, 2005.
H. Elman, A. Ramage, and D. Silvester, Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33, 2-14, 2007.
C. A. J. Fletcher, Computational Techniques for Fluid Dynamics, Springer-Verlag, 1988.
M. Lee and R. D. Moser, Direct numerical simulation of turbulent channel flow up to Re ≈ 5200, J. Fluid Mech. (2015), vol 774, pp. 395-415. Doi 10.1017/jfm.2015.268.
A. J. Smits and I. Marusic, Wall-bounded turbulence. Phys. Today 66 (9), 25-30, 2013.
D. Silvester, H. Elman and A. Ramage, Incompressible Flow and Iterative Solver Software (IFISS) version 3.5, September, 2016, http://www.manchester.ac.uk/ifiss/
Valerla Simoncini, and Daniel B. Szyld, Theory of inexact Krylov subspace methods and applications to scientific computing, SIAM J. Sci. Comput, 25 (2003), No. 2, pp 454-477.