非定常不可压Navier-Stokes方程基于Crank-Nicolson格式的两水平变分多尺度方法
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摘要
不可压缩粘性流是密度不发生变化的流体运动.它们被用来描述许多重要的物理现象,例如:天气、洋流、绕翼型流动和动脉内的血液流动.Navier-Stokes方程是不可压缩粘性流的基本方程.因此,求解Navier-Stokes方程的数值方法在近几十年得到了广泛的关注.本文主要给出非定常不可压Navier-Stokes方程基于Crank-Nicolson格式的两水平变分多尺度方法.该方法分为两步:第一步,在粗网格上求解稳定的非线性Navier-Stokes系统;第二步,在细网格上求解稳定的线性问题去校正粗网格上的解.通过该方法推导的速度的误差估计关于时间是二阶收敛的.数值实验验证了在粗细网格匹配合理的情形下,本文的方法与直接在细网格上使用单网格的变分多尺度方法相比,可以节约大量的计算时间.
The incompressible viscous flows are fluid movements that do not change in density.They are used to describe many important physical phenomena such as weather, ocean currents,flow around airfoil, and blood flow within the arteries. The Navier-Stokes equations are the basic equations for incompressible viscous flows. Therefore, the numerical method for solving Navier-Stokes equations has been paid more and more attention in recent decades. In this paper, we mainly study a two-level fully discrete finite element variational multiscale method based on Crank-Nicolson scheme for the unsteady Navier-Stokes equations. The method is carried out in two steps. A stabilized nonlinear Navier-Stokes system is solved on a coarse grid at the first step, and the second step is that a stabilized linear problem is solved on a fine grid to correct the coarse grid solution. Error estimate of the velocity which is derived via the two-level finite element variational multiscale method is of second-order in time. Numerical experiments show that the method of this paper can save a lot of computation time compared with the finite element variational method which uses a one-level grid directly on the fine grid in the case of coarse grid matching.
The incompressible viscous flows are fluid movements that do not change in density.They are used to describe many important physical phenomena such as weather, ocean currents,flow around airfoil, and blood flow within the arteries. The Navier-Stokes equations are the basic equations for incompressible viscous flows. Therefore, the numerical method for solving Navier-Stokes equations has been paid more and more attention in recent decades. In this paper, we mainly study a two-level fully discrete finite element variational multiscale method based on Crank-Nicolson scheme for the unsteady Navier-Stokes equations. The method is carried out in two steps. A stabilized nonlinear Navier-Stokes system is solved on a coarse grid at the first step, and the second step is that a stabilized linear problem is solved on a fine grid to correct the coarse grid solution. Error estimate of the velocity which is derived via the two-level finite element variational multiscale method is of second-order in time. Numerical experiments show that the method of this paper can save a lot of computation time compared with the finite element variational method which uses a one-level grid directly on the fine grid in the case of coarse grid matching.
引文
[1]Temam R.Navier-Stokes Equations:Theory and Numerical Analysis[M].Amsterdam:North-Holland Publishing Company,1984
[2]Girault V,Raviart P A.Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms[M].Berlin Heidelberg:Springer-Verlag,1986
[3]Wang K.A new defect correction method for the Navier-Stokes equations at high Reynolds numbers[J].Applied Mathematics and Computation,2010,216(11):3252-3264
[4]Liu Q F,Hou Y R.A two-level defect-correction method for Navier-Stokes equations[J].Bulletin of the Australian Mathematical Society,2010,81(3):442-454
[5]Zhang Y,He Y N.Assessment of subgrid-scale models for the incompressible Navier-Stokes equations[J].Journal of Computational and Applied Mathematics,2010,234(2):593-604
[6]Shang Y Q.A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations[J].Journal of Computational Physics,2013,233(1):210-226
[7]John V,Kaya S.A finite element variational multiscale method for the Navier-Stokes equations[J].SIAMJournal on Scientific Computing,2005,26(5):1485-1503
[8]Li J,He Y N.A stabilized finite element method based on two local Gauss integrations for the Stokes equations[J].Journal of Computational and Applied Mathematics,2008,214(1):58-65
[9]He Y N,Li J.A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations[J].Applied Numerical Mathematics,2008,58(10):1503-1514
[10]Zheng H B,Hou Y R,Shi F,et al.A finite element variational multiscale method for incompressible flows based on two local Gauss integrations[J].Journal of Computational Physics,2009,228(16):5961-5977
[11]He Y N,Sun W W.Stability and convergence of the Crank-Nicolson/Adams-Bsahforth scheme for the time-dependent Navier-Stokes equations[J].SIAM Journal on Numerical Analysis,2007,45(2):837-869
[12]He Y N.The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data[J].Mathematics of Computation,2008,77(264):2097-2124
[13]He Y N.Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations[J].SIAM Journal on Numerical Analysis,2003,41(4):1263-1285
[14]He Y N,Liu K M.A multi-level finite element method for the time-dependent Navier-Stokes equations[J].Numerical Methods for Partial Differential Equations,2005,21(6):1052-1068
[15]Shang Y Q.Error analysis of a fully discrete finite element variational multiscale method for time-dependent incompressible Navier-Stokes equations[J].Numerical Methods for Partial Differential Equations,2013,29(6):2025-2046
[16]Heywood J G,Rannacher R.Finite element approximation of the nonstationary Navier-Stokes problem IV:error analysis for second-order time discretization[J].SIAM Journal on Numerical Analysis,1990,27(2):353-384
[17]Hecht F.New development in FreeFem++[J].Journal of Numerical Mathematics,2012,20(3-4):251-265
[2]Girault V,Raviart P A.Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms[M].Berlin Heidelberg:Springer-Verlag,1986
[3]Wang K.A new defect correction method for the Navier-Stokes equations at high Reynolds numbers[J].Applied Mathematics and Computation,2010,216(11):3252-3264
[4]Liu Q F,Hou Y R.A two-level defect-correction method for Navier-Stokes equations[J].Bulletin of the Australian Mathematical Society,2010,81(3):442-454
[5]Zhang Y,He Y N.Assessment of subgrid-scale models for the incompressible Navier-Stokes equations[J].Journal of Computational and Applied Mathematics,2010,234(2):593-604
[6]Shang Y Q.A two-level subgrid stabilized Oseen iterative method for the steady Navier-Stokes equations[J].Journal of Computational Physics,2013,233(1):210-226
[7]John V,Kaya S.A finite element variational multiscale method for the Navier-Stokes equations[J].SIAMJournal on Scientific Computing,2005,26(5):1485-1503
[8]Li J,He Y N.A stabilized finite element method based on two local Gauss integrations for the Stokes equations[J].Journal of Computational and Applied Mathematics,2008,214(1):58-65
[9]He Y N,Li J.A stabilized finite element method based on local polynomial pressure projection for the stationary Navier-Stokes equations[J].Applied Numerical Mathematics,2008,58(10):1503-1514
[10]Zheng H B,Hou Y R,Shi F,et al.A finite element variational multiscale method for incompressible flows based on two local Gauss integrations[J].Journal of Computational Physics,2009,228(16):5961-5977
[11]He Y N,Sun W W.Stability and convergence of the Crank-Nicolson/Adams-Bsahforth scheme for the time-dependent Navier-Stokes equations[J].SIAM Journal on Numerical Analysis,2007,45(2):837-869
[12]He Y N.The Euler implicit/explicit scheme for the 2D time-dependent Navier-Stokes equations with smooth or non-smooth initial data[J].Mathematics of Computation,2008,77(264):2097-2124
[13]He Y N.Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations[J].SIAM Journal on Numerical Analysis,2003,41(4):1263-1285
[14]He Y N,Liu K M.A multi-level finite element method for the time-dependent Navier-Stokes equations[J].Numerical Methods for Partial Differential Equations,2005,21(6):1052-1068
[15]Shang Y Q.Error analysis of a fully discrete finite element variational multiscale method for time-dependent incompressible Navier-Stokes equations[J].Numerical Methods for Partial Differential Equations,2013,29(6):2025-2046
[16]Heywood J G,Rannacher R.Finite element approximation of the nonstationary Navier-Stokes problem IV:error analysis for second-order time discretization[J].SIAM Journal on Numerical Analysis,1990,27(2):353-384
[17]Hecht F.New development in FreeFem++[J].Journal of Numerical Mathematics,2012,20(3-4):251-265