A classification of the second order projection methods to solve the Navier-Stokes equations
Many projection methods have been progressively con- structed to find more accurate and efficient solution of the Navier- Stokes equations. In this paper, we consider most recently con- structed projection methods: the pressure correction method, the gauge method, the consistent splitting method, the Gauge-Uzawa method, and the stabilized Gauge-Uzawa method. Each method has different background and theoretical proof. We prove equivalentness of the pressure correction method and the stabilized Gauge-Uzawa method. Also we will obtain that the Gauge-Uzawa method is equiv- alent to the gauge method and the consistent splitting method. We gather theoretical results of them and conclude that the results are also valid on other equivalent methods.
Subject classification65M12, 65M15, 76D05
Sponsor(s)This study was supported by 2014 Research Grant from Kangwon National University (No. 120140361).
A.J. Chorin, Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968), 745–762. (Google Scholar)
W. E and J.-G. Liu, Gauge method for viscous incompressible flows, Comm. Math. Sci. (2003) 317–332. (Google Scholar)
V. Girault, and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations, Springer-Verlag (1986). (Google Scholar)
J.L. Guermond and J. Shen, On the error estimates of rotational pressure- correction projection methods, Math. Comp. 73 (2004), 1719–1737. (Google Scholar)
J.L. Guermond and J. Shen, A new class of truly consistent splitting schemes for incompressible flows, J. Comput. Phys. 192 (2003), 262–276. (Google Scholar)
R.H. Nochetto and J.-H. Pyo, Error estimates for semi-discrete gauge methods for the Navier-Stokes equations, Math. Comp. 74 (2005), 521–542. (Google Scholar)
R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part I : the Navier-Stokes equations, SIAM J. Numer. Anal. 43 (2005), 1043–1068. (Google Scholar)
R.H. Nochetto and J.-H. Pyo, A finite element Gauge-Uzawa method. Part II : Boussinesq equations, Math. Models Methods Appl. Sci. 16 (2006), 1599–1626. (Google Scholar)
J.-H. Pyo, An overview of BDF2 Gauge-Uzawa methods for incompressible flows, KSIAM 15 (2011), 233–251. (Google Scholar)
J.-H. Pyo, Error estimates for the second order semi-discrete stabilized Gauge-Uzawa method for the Navier-Stokes equations, IJNAM 10 (2013), 24–41. (Google Scholar)
J.-H. Pyo and J. Shen, Normal mode analysis of second-order projection methods for incompressible flows, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), 817–840. (Google Scholar)
J.-H. Pyo and J. Shen, Gauge Uzawa methods for incompressible flows with variable density, J. Comput. Phys. 211 (2007), 181–197. (Google Scholar)
R. Temam, Sur l’approximation de la solution des equations de Navier-Stokes par la methode des pas fractionnaires. II, Arch. Rational Mech. Anal. 33 (1969), 377–385. (Google Scholar)
R. Temam, Navier-Stokes Equations, AMS Chelsea Publishing, (2001). (Google Scholar)
L.J.P. Timmermanns, P.D. Minev, and F.N. Van De Vosse, An approximate projection scheme for incompressible flow using spectral elements, Int. J. Num. Meth. Fluids 22 (1996), 673–688. (Google Scholar)
- There are currently no refbacks.
ISSN: 1976-8605 (Print), 2288-1433 (Online)
Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: email@example.com