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An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations
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{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 = \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 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
Institute for Computational Mechanics in Propulsion (ICOMP)
The Institute for Computational Mechanics in Propulsion (ICOMP) is a combined activity of Case Western Reserve University, Ohio Aerospace Institute (OAI) and NASA Lewis. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1991 are described
On high-order pressure-robust space discretisations, their advantages for incompressible high Reynolds number generalised Beltrami flows and beyond
An improved understanding of the divergence-free constraint for the
incompressible Navier--Stokes equations leads to the observation that a
semi-norm and corresponding equivalence classes of forces are fundamental for
their nonlinear dynamics. The recent concept of {\em pressure-robustness}
allows to distinguish between space discretisations that discretise these
equivalence classes appropriately or not. This contribution compares the
accuracy of pressure-robust and non-pressure-robust space discretisations for
transient high Reynolds number flows, starting from the observation that in
generalised Beltrami flows the nonlinear convection term is balanced by a
strong pressure gradient. Then, pressure-robust methods are shown to outperform
comparable non-pressure-robust space discretisations. Indeed, pressure-robust
methods of formal order are comparably accurate than non-pressure-robust
methods of formal order on coarse meshes. Investigating the material
derivative of incompressible Euler flows, it is conjectured that strong
pressure gradients are typical for non-trivial high Reynolds number flows.
Connections to vortex-dominated flows are established. Thus,
pressure-robustness appears to be a prerequisite for accurate incompressible
flow solvers at high Reynolds numbers. The arguments are supported by numerical
analysis and numerical experiments.Comment: 43 pages, 18 figures, 2 table
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