AI 中文总结
针对不可压缩Navier-Stokes方程，提出基于IMEX-BDFk的全离散有限元格式，显式处理非线性对流项，隐式处理线性Stokes部分。建立稳定性、一致有界性及最优阶误差估计，通过数值实验验证格式有效性与理论收敛率。
AI 中文摘要
本文在全离散有限元框架下，针对具有无滑移边界条件的不可压缩Navier-Stokes方程提出并分析了一类高阶数值格式。时间离散采用k阶（k = 1，...，6）隐显向后差分公式（IMEX-BDFk），非线性对流项显式处理，线性Stokes部分隐式处理，空间离散利用Taylor-Hood有限元。建立了数值解的稳定性和一致有界性，在无CFL型条件下，即时间步长与空间网格大小无关时，进一步建立了时空最优阶误差估计。三维情况下给出了速度的L2和H1范数误差估计以及压力的L2范数误差估计，所有变量的时间收敛率高达六阶。通过数值实验证明了该格式的有效性并验证了理论收敛率。
英文摘要
In this paper, we propose and analyze a class of high-order numerical schemes within a fully discrete finite element framework for the incompressible Navier-Stokes equations with no-slip boundary conditions. The temporal discretization employs a kth-order (k=1,...,6) implicit-explicit backward difference formula (IMEX-BDFk), in which the nonlinear convection term is treated explicitly and the linear Stokes part implicitly, whereas the spatial discretization utilizes Taylor-Hood finite elements. We establish the stability and uniform boundedness of the numerical solution. We further establish optimal order error estimates in both space and time without any CFL-type condition, in the sense that the time step is independent of the spatial mesh size. In three dimensions, these include L2- and H1-norm error estimates for the velocity and L2-norm error estimates for the pressure, with temporal convergence rates up to sixth order for all variables. Numerical experiments are presented to demonstrate the effectiveness of the scheme and to confirm the theoretical convergence rates.
Comments42pages, 36figures, 37references
