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An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations

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Abstract

We investigate the long time behavior of the following efficient second-order in time scheme for the 2D Navier–Stokes equations in a periodic box:

$$\begin{array}{ll}{\frac{3\omega^{n+1} - 4\omega^n + \omega^{n-1}}{2k} + \nabla^\perp(2\psi^n - \psi^{n-1}) \cdot \nabla(2\omega^n - \omega^{n-1})- \nu\Delta\omega^{n+1} = f^{n+1},} \\ {\quad -{\Delta} {\psi}^{n} = {\omega}^{n}.}\end{array}$$

The scheme is a combination of a 2nd-order in time backward-differentiation and a particular explicit Adams–Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson solver is needed at each time step. We prove uniform in time bounds on this scheme in \({{\dot{L}^2,\, \dot{H}^1_{per}}}\) 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.

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  1. Department of Mathematics, Florida State University, Tallahassee, FL, 32306, USA

    Xiaoming Wang

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Wang, X. An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations. Numer. Math. 121, 753–779 (2012). https://doi.org/10.1007/s00211-012-0450-3

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