Navier-Stokes方程的二阶解耦BDF格式的稳定性和收敛性分析  

Stability and Convergence Analysis of Second-order BDF Decoupled Scheme for Navier-Stokes Equations

作　　者：牟小凤 夏泽宇 李茂军 MOU Xiao-feng;XIA Ze-yu;LI Mao-jun(School of Mathematical Sciences,University of Electronic Science and Technology of China,Chengdu Sichuan 611731,China)

机构地区：[1]电子科技大学数学科学学院,成都611731

出　　处：《西华师范大学学报（自然科学版）》2023年第1期32-41,共10页Journal of China West Normal University(Natural Sciences)

基　　金：国家自然科学基金项目(11871139)。

摘　　要：利用有限元(FEM)的空间离散和二阶向后差分公式(BDF)的时间离散,针对不可压Navier-Stokes方程设计了一种解耦格式。“解耦”技术是指在速度场方程中引入中间变量并对压力项进行显式处理,从而将速度场方程变为对称正定系统,再通过额外求解泊松方程得到压力项。通过添加修正项,得到了该数值格式的无条件能量稳定性,进而得到了其格式的唯一可解性。针对不可压Navier-Stokes方程的解耦格式,本文首次通过引入中间映射项对其进行了最优收敛估计。最后通过数值算例验证了该数值格式的精度和稳定性。This paper proposes a decoupled scheme for solving incompressible Navier-Stokes equations by a combination of finite element method(FEM) for spatial discretization and second-order backward differential formulation(BDF) for temporal discretization.The “decoupled” technique turns the flow equation into a symmetric positive definite system by introducing an intermediate variable and treating the pressure function explicitly, and then computes the pressure by solving a Poission-type equation.The unconditional energy stability and unique solvability of the numerical scheme are obtained by adding the correction term.It is the first time for the paper to get the optimal error estimates of decoupled scheme by introducing an intermediate variable for solving incompressible Navier-Stokes equations.In the end, several numerical examples are presented to verify the accuracy and stability of the numerical scheme.

关 键 词：NAVIER-STOKES方程 有限元方法 解耦BDF 无条件稳定 唯一可解性 最优误差估计 

分 类 号：O241.82[理学—计算数学]