摘要
在连续解的正则性假设条件下,基于亚格子稳定模型和算子分裂方法提出了非定常不可压Navier-Stokes方程的有限元算子分裂算法.其主要思想是:利用算子分裂方法把非线性项和不可压缩项分开,首先求解一个线性化的Burger's问题得到有限元解■,然后再求解一个Stokes问题得到解u■.证明了速度的误差估计关于时间是一阶收敛的,并给出数值实验验证了理论的正确性.
Under the regularity assumptions on the continuous solution, we provide a finite element operator splitting method for the simulation of unsteady incompressible Navier-Stokes equations, which is based on the subgrid model. It is a two-step scheme in which the nonlinearity and incompressibility are split into different steps. First, a linear Burger's system is solved, and the solution of the finite element ■ is obtained. Then a Stokes problem is solved, and its solution u■ is obtained. We derive the error bound of the approximate velocity which is first-order in time. Numerical experiments have verified the correctness of the theoretical analysis.
引文
[1]CHORIN A J.Numerical Solution of the Navier-Stokes Equations[J].Computational Fluid Mechanics,1968,22(104):745-762.
[2]GIRAULT V,RAVIART P A.Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms[M]//Finite Element Methods for Navier-Stokes Equations:Theory and Algorithms.New York:Springer-Verlag,1986.
[3]GLOWINSKI R.Finite Element Methods for Incompressible Viscous Flow[J].Handbook of Numerical Analysis,2003,9:3-1176.
[4]BLASCO J,CODINA R.Error Estimates for an Operator-Splitting Method for Incompressible Flows[J].Applied Numerical Mathematics,2004,51(1):1-17.
[5]QUARTERONI A,SALERI F,VENEZIANI A.Factorization Methods for the Numerical Approximation of NavierStokes Equations[J].Computer Method in Applied Mechanics and Engineering,2000,188(1-3):505-526.
[6]SHEN J.On Error Estimates of Projection Methods for Navier-Stokes Equations:First-Order Schemes[J].SIAM Journal on Numerical Analysis,1992,29(1):57-77.
[7]GUERMOND J L,QUARTAPELLE L.On the Approximation of the Unsteady Navier-Stokes Equations by Finite Element Projection Methods[J].Numerische Mathematik,1998,80(2):207-238.
[8]BLASCO J,CODINA R,HUERTA A.A Fractional-Step Method for the Incompressible Navier-Stokes Equations Related to a Predictor-Multicorrector Algorithm[J].International Journal for Numerical Methods in Fluids,2015,28(10):1391-1419.
[9]KIM J,MOIN P.Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations[J].Journal of Computational Physics,1985,59(2):308-323.
[10]DONEA J,GIULIANI S,LAVAL H,et al.Finite Element Solution of the Unsteady Navier-Stokes Equations by a Fractional Step Method[J].Computer Methods in Applied Mechanics and Engineering,1982,30(1):53-73.
[11]HECHT F.New Development in Freefem++[J].Journal of Numerical Mathematics,2012,20(3-4):251-266.
[12]杨晓成,尚月强.Navier-Stokes方程的回溯两水平有限元变分度尺度方法[J].西南大学学报(自然科学版),2017,39(10):47-57.