Abstract
We consider the second-order projection schemes for the time-dependent natural convection problem. By the projection method, the natural convection problem is decoupled into two linear subproblems, and each subproblem is solved more easily than the original one. The error analysis is accomplished by interpreting the second-order time discretization of a perturbed system which approximates the time-dependent natural convection problem, and the rigorous error analysis of the projection schemes is presented. Our main results of the second order projection schemes for the time-dependent natural convection problem are that the convergence for the velocity and temperature are strongly second order in time while that for the pressure is strongly first order in time.
Similar content being viewed by others
References
R. Araya, G.R. Barrenechea, A. H. Poza, F. Valentin: Convergence analysis of a residual local projection finite element method for the Navier-Stokes equations. SIAM J. Numer. Anal. 50 (2012), 669–699.
G. Chen, M. Feng, H. Zhou: Local projection stabilized method on unsteady Navier-Stokes equations with high Reynolds number using equal order interpolation. Appl. Math. Comput. 243 (2014), 465–481.
A. J. Chorin: Numerical solution of the Navier-Stokes equations. Math. Comput. 22 (1968), 745–762.
A. Cibik, S. Kaya: A projection-based stabilized finite element method for steady-state natural convection problem. J. Math. Anal. Appl. 381 (2011), 469–484.
B. Du, H. Su, X. Feng: Two-level variational multiscale method based on the decoupling approach for the natural convection problem. Int. Commun. Heat. Mass. 61 (2015), 128–139.
Y. He: Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations. SIAM J. Numer. Anal. 41 (2003), 1263–1285.
Y. He: Stability and error analysis for a spectral Galerkin method for the Navier-Stokes equations with H 2 or H 1 initial data. Numer. Methods Partial Differ. Equations 21 (2005), 875–904.
J.G. Heywood, R. Rannacher: Finite element approximation of the nonstationary Navier-Stokes problem. I. Regularity of solutions and second-order error estimates for spatial discretization. SIAM J. Numer. Anal. 19 (1982), 275–311.
M.T. Manzari: An explicit finite element algorithm for convection heat transfer problems. Int. J. Numer. Methods Heat Fluid Flow 9 (1999), 860–877.
J. H. Pyo: A classification of the second order projection methods to solve the Navier-Stokes equations. Korean J. Math. 22 (2014), 645–658.
Y. X. ian, T. Zhang: On error estimates of the projection method for the timedependent natural convection problem: first order scheme. Submitted to Comput. Math. Appl.
Y. X. Qian, T. Zhang: On error estimates of a higher projection method for the timedependent natural convection problem. Submitted to Front. Math. China.
J. Shen: On error estimates of projection methods for Navier-Stokes equations: Firstorder schemes. SIAM J. Numer. Anal. 29 (1992), 57–77.
J. Shen: On error estimates of some higher order projection and penalty-projection methods for Navier-Stokes equations. Numer. Math. 62 (1992), 49–73.
S. Shen: The finite element analysis for the conduction-convection problems. Math. Numer. Sin. 16 (1994), 170–182. (In Chinese.)
J. Shen: On error estimates of the projection methods for the Navier-Stokes equations: Second-order schemes. Math. Comput. 65 (1996), 1039–1065.
M. Tabata, D. Tagami: Error estimates of finite element methods for nonstationary thermal convection problems with temperature-dependent coefficients. Numer. Math. 100 (2005), 351–372.
R. Témam: Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II. Arch. Ration. Mech. Anal. 33 (1969), 377–385. (In French.)
R. Témam: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications, Vol. 2, North-Holland, Amsterdam, 1984.
A.W. Vreman: The projection method for the incompressible Navier-Stokes equations: the pressure near a no-slip wall. J. Comput. Phys. 263 (2014), 353–374.
Y. Zhang, Y. Hou, J. Zhao: Error analysis of a fully discrete finite element variational multiscale method for the natural convection problem. Comput. Math. Appl. 68 (2014), 543–567.
T. Zhang, Z. Tao: Decoupled scheme for time-dependent natural convection problem II: time semidiscreteness. Math. Probl. Eng. 2014 (2014), Article ID 726249, 23 pages.
T. Zhang, J. Yuan, Z. Si: Decoupled two-grid finite element method for the timedependent natural convection problem I: Spatial discretization. Numer. Methods Partial Differ. Equations 31 (2015), 2135–2168.
X. Zhang, P. Zhang: Meshless modeling of natural convection problems in nonrectangular cavity using the variational multiscale element free Galerkin method. Eng. Anal. Bound. Elem. 61 (2015), 287–300.
T. Zhang, X. Zhao, P. Huang: Decoupled two level finite element methods for the steady natural convection problem. Numer. Algorithms 68 (2015), 837–866.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research has been supported by CAPES and CNPq of Brazil (No. 88881.068004/2014.01), the NSF of China (No. 11301157), and the FDYS of Henan Polytechnic University (J2015-05).
Rights and permissions
About this article
Cite this article
Qian, Y., Zhang, T. The second order projection method in time for the time-dependent natural convection problem. Appl Math 61, 299–315 (2016). https://doi.org/10.1007/s10492-016-0133-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-016-0133-y