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应用数学进展  >> Vol. 1 No. 2 (November 2012)

二维非定常Navier-Stokes方程的Euler隐/显格式子格涡旋粘性非协调有限元法
The Euler Implicit/Explicit Schemes with Nonconforming Finite Element Method of Subgrid Eddy Viscosity Type for the 2D Time-Dependent Navier-Stokes Equations

DOI: 10.12677/aam.2012.12008, PDF, HTML, 下载: 3,154  浏览: 13,788  国家自然科学基金支持

作者: 郭英文, 冯民富：四川大学数学学院，成都

关键词: Euler隐/显格式；子格涡旋粘性法；非协调C-R元； Euler Implicit/Explicit Schemes; Subgrid Eddy ViscosityMethod;Nonconforming C-R Element

摘要:

本文研究了二维高雷诺数情形下，非定常不可压Navier-Stokes方程的Euler隐/显格式子格涡旋粘性非协调有限元方法。隐式处理线性项，避免了时间步长的苛刻限制，显式处理非线性项，使得对所有时间层求解时，系统矩阵为同一常数矩阵；时间项做向后Euler差分离散，空间用C-R非协调有限元逼近，构造子格涡旋粘性有限元方法，克服了在情形下Galerkin有限元方法的不稳定现象。本文改善了稳定性下对时间步长的限制，并给出了不依赖粘性系数的速度和压力误差估计。

Abstract: In this paper, an Euler implicit/explicit scheme with nonconforming finite element method of subgrid eddy viscosity type for solving the 2D nonstationary incompressible Navier-Stokes equations under high Reynolds number  is considered. The implicit/explicit scheme which is implicit for the linear terms and explicit for the nonlinear term, avoids the severely restricted time step size from stability requirement and results in a linear system with a same constant matrix at each level of time. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation is used for the velocity and pressure field with the subgrid eddy viscosity technique, to cope with usual instabilities caused by Galerkin finite element methods. This paper also improved the restricted time step size which under stable conditions and given error estimates of velocity and pressure which independent on the viscosity .

 

文章引用: 郭英文, 冯民富. 二维非定常Navier-Stokes方程的Euler隐/显格式子格涡旋粘性非协调有限元法[J]. 应用数学进展, 2012, 1(2): 59-70. http://dx.doi.org/10.12677/aam.2012.12008

参考文献

[1] Y. N. He. The euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data. Mathematics of Computation, 2008, 77(264): 2097-2124.
[2] L. G. Davis, F. Pahlevani. Semi-implicit schemes for transient Navier-Stokes equations and eddy viscosity models. Wiley InterScience, 2007: 1-20.
[3] J. Li, Y. He and Z. Chen. A new stabilized finite element method for the transient Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 2007, 197(1-4): 22-35.
[4] R. Temam. Navier-Stokes equations and nonlinear functional analysis. 2nd Edition, Philadelphia: SIAM, 1995.
[5] W. Layton, L. Tobiska. A two-level method with backtrackking for the Navier-Stokes equations. SIAM Journal on Numerical Analysis, 1998, 35(5): 2035-2054.
[6] V. Girault, P. A. Raviart. Finite element method for Navier-Stokes equations: Theory and algorithms. Berlin, Heidelberg: Springer-Verlag, 1987.
[7] M. Crouzeix, P. Raviart. Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. ESAIM: Mathematical Modelling and Numerical Analysis, 1973, 7(R3): 33-75.
[8] A. Quarteroni, A. Valli. Numerical approximation of partial differential equations. Berlin: Springer-Verlag, 1997.
[9] T. J. R. Hughes, A. N. Brooks. A multi-dimensional up-wind scheme with no crosswind Diffusion. In: T. J. R. Hughes, Ed., Finite element methods for convection domination flows. ASME Monograph AMD-34, 1979: 19-35.
[10] J.-L. Guermond. Stabilization of Galerkin approximation of transport equations by subgrid modeling. Mathematical Modelling and Numerical Analysis, 1999, 33(6): 1293-1316.
[11] W. Layton. A connection between subgrid scale eddy viscosity and mixed methods. Applied Mathematics and Computation, 2002, 133(1): 147-157.
[12] V. John, S. Kaya. A finite element variational multiscale method for the Navier-Stokes equations. SIAM Journal on Scientific Computing, 2005, 26(5): 1485-1503.
[13] V. John, S. Kaya. Finite element error analysis of a variational multiscale method for the Navier-Stokes equations. Advances in Computational Mathematics, 2008, 28: 43-61.
[14] S. Kaya, B. Riviere. A discontinuous subgrid eddy viscosity method for the time-dependent Navier-Stokes equations. SIAM Journal on Numerical Analysis, 2005, 43(4): 1572-1595.
[15] S. Kaya, W. Layton. Subgrid-scale eddy viscosity methods are variational multiscale method. Tech. Report TR-MATH 03-05, University of Pittsburgh, 2003.
[16] J. G. Heywood, R. Rannacher. Finite-element approximations of the nonstationary Navier-Stokes problem, Part I: Regularity of solutions and second-order spatial discretization. SIAM Journal on Numerical Analysis, 1982, 19(2): 275-311.
[17] R. Temam. Navier-Stokes equations: Theory and numerical analysis. 3rd Edition, Amsterdam: North-Holland, 1983.
[18] 王烈衡, 许学军. 有限元方法的数学基础[M]. 北京: 科学出版社, 2004.
[19] V. John, J. Maubach and L. Tobiska. Nonconforming streamline-diffusion-finite-element-methods for convection-diffusion problems. Numerische Mathematik, 1997, 78: 165-188.
[20] Z. Cai, J. Douglas Jr. and X. Ye. A stable nonconforming quadric-lateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo, 1999, 36: 215-232.