二维Navier-Stokes方程经典高效格式的长时间稳定性
Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations
AI总结：
针对二维不可压缩Navier-Stokes方程，证明了一种隐式处理粘性项、显式处理非线性对流项的经典隐式-显式格式在周期边界条件下时间步长足够小时具有长时间稳定性，进而导致全局吸引子和不变测度收敛。
摘要：
我们证明，对于二维不可压缩Navier-Stokes方程，一种流行的经典隐式-显式格式（隐式处理粘性项，显式处理非线性对流项）在周期边界条件下，若时间步长足够小，则具有长时间稳定性。$L^2$和$H^1$范数下的长时间稳定性进一步导致该格式的全局吸引子和不变测度在时间步长趋于零时收敛到NSE本身的全局吸引子和不变测度。我们考虑了时间半离散格式以及采用Galerkin Fourier谱或配置Fourier谱方法的全离散格式。
英文摘要：
We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier--Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. The long time stability in the $L^2$ and $H^1$ norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the NSE itself at vanishing time step. Both semi-discrete in time and fully discrete schemes with either Galerkin Fourier spectral or collocation Fourier spectral methods are considered.
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