一种高效的二阶时间格式用于逼近二维Navier-Stokes方程的长时间统计性质
An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations
AI总结：
提出一种结合二阶BDF和显式Adams-Bashforth处理对流项的时间格式，证明其长时间均匀有界性，并收敛到NSE的统计性质。
摘要：
我们研究以下高效二阶时间格式在周期域中二维Navier-Stokes方程的长时间行为：$$ \frac{3ω^{n+1}-4ω^n+ω^{n-1}}{2k} + \nabla^\perp(2ψ^n-ψ^{n-1})\cdot\nabla(2ω^n-ω^{n-1}) - νΔω^{n+1} = f^{n+1}, \quad -Δψ^n = \om^n. $$ 该格式结合了二阶时间向后微分（BDF）和对流项的特殊显式Adams-Bashforth处理。因此每个时间步只需求解一个线性常系数Poisson型问题。我们证明了在时间步长足够小的条件下，该格式在$\dL2$、$\dH1$和$\dot{H}^2_{per}$中的时间一致有界性。这些时间一致估计进一步导致当时间步长趋近于零时，格式的长时间统计（平稳统计性质）收敛到NSE本身的统计性质。还讨论了采用Galerkin Fourier或配点Fourier谱方法的全离散格式。
英文摘要：
We investigate the long tim behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box: $$ \frac{3ω^{n+1}-4ω^n+ω^{n-1}}{2k} + \nabla^\perp(2ψ^n-ψ^{n-1})\cdot\nabla(2ω^n-ω^{n-1}) - νΔω^{n+1} = f^{n+1}, \quad -Δψ^n = \om^n. $$ The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme in $\dL2$, $\dH1$ and $\dot{H}^2_{per}$ provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed.
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