Korean J. Math.  Vol 27, No 1 (2019)  pp.193-219
DOI: https://doi.org/10.11568/kjm.2019.27.1.193

The second-order stabilized Gauge-Uzawa method for incompressible flows with variable density

Taek-cheol Kim, Jae-Hong Pyo

Abstract


The Navier-Stokes equations with variable density are challenging problems in numerical analysis community.   We recently built the 2nd order stabilized Gauge-Uzawa method [SGUM] to solve the Navier-Stokes equations   with constant density and have estimated theoretically optimal accuracy. Also we proved that SGUM is unconditionally stable. In this paper, we apply SGUM to the Navier-Stokes equations with nonconstant variable density   and find out the stability condition of the algorithms. Because the condition is rather strong to apply to real problems, we consider Allen-Cahn scheme to construct unconditionally stable scheme. 


Keywords


Gauge-Uzawa method, finite element method, pressure correction method, Allen-Cahn

Subject classification

65M12, 65M60, 76D05

Sponsor(s)

This study was supported by 2016 Research Grant from Kangwon National University (No.520160376).

Full Text:

PDF

References


S.M. Allen and J.W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Ac ta Metallurgica, 27 (1979), 1085–1095. (Google Scholar)

Alexandre Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Scioences, vol. 159, Springer, New York, 2004. (Google Scholar)

J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows, J. comput. Phys, 165 (2000), 167–188. (Google Scholar)

J.-.L. Guermond and A. Salgado, A spliiting method for incompressible flows with variable density based on a pressure Poisson equation, J. comput. Phys, 228 (2009), 2834–2846 (Google Scholar)

J.-.L. Guermond and A. Salgado, Error analysis of a fractional time-stepping technique for incompressible flows with variable density, SIAM J. Numer. Anal., 49 (2011), 917–944. (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)

F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), 251– 265. (Google Scholar)

Claes Johnson, Numerical solution of partial differntial equations by the finite element method, Cambridge University Press, Cambridge, 1987. (Google Scholar)

J. Kim, Y. Li, H.G. Lee and D. Jeong, An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation, Comput. Math. Appl. 60 (2010), 1591–1606. (Google Scholar)

J.C. Latch ́e and K. Saleh, A convergnet staggered scheme for the variable density incompressible Navier-Stokes equations, Math. Comp. accepted for publication (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, Error estimates for the second order semi-discrete stabilized gauge- Uzawa method for the Navier-Stokes equations, Inter. J. Numer. Anal. & Model. 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 221 (2007), 184–197. (Google Scholar)

J. Shen, X. Yang, J. J. Feng and C. Liu, Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys 218 (2006), 417–428. (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)

Gr ́etar Tryggvason, Numerical simulations of the Ralyleigh-Taylor instability, J. Comput. Phys 75 (1988), 253–282. (Google Scholar)


Refbacks

  • 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: kkms@kangwon.ac.kr