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

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 .