The second-order stabilized Gauge-Uzawa method for incompressible flows with variable density
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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.
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[1] 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
[2] Alexandre Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Scioences, vol. 159, Springer, New York, 2004. Google Scholar
[3] J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows, J. comput. Phys, 165 (2000), 167–188. Google Scholar
[4] 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
[5] 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
[6] J.L. Guermond and J. Shen, On the error estimates of rotational pressure- correction projection methods, Math. Comp. 73 (2004), 1719–1737. Google Scholar
[7] F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), 251– 265. Google Scholar
[8] Claes Johnson, Numerical solution of partial differntial equations by the finite element method, Cambridge University Press, Cambridge, 1987. Google Scholar
[9] 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
[10] 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
[11] 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
[12] 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
[13] 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
[14] 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
[15] J.-H. Pyo and J. Shen, Gauge-Uzawa methods for incompressible flows with variable density, J. Comput. Phys 221 (2007), 184–197. Google Scholar
[16] 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
[17] R. Temam, Navier-Stokes equations, AMS Chelsea Publishing, 2001. Google Scholar
[18] 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
[19] Gr etar Tryggvason, Numerical simulations of the Ralyleigh-Taylor instability, J. Comput. Phys 75 (1988), 253-282. Google Scholar