703 research outputs found
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
On the fourth-order accurate compact ADI scheme for solving the unsteady Nonlinear Coupled Burgers' Equations
The two-dimensional unsteady coupled Burgers' equations with moderate to
severe gradients, are solved numerically using higher-order accurate finite
difference schemes; namely the fourth-order accurate compact ADI scheme, and
the fourth-order accurate Du Fort Frankel scheme. The question of numerical
stability and convergence are presented. Comparisons are made between the
present schemes in terms of accuracy and computational efficiency for solving
problems with severe internal and boundary gradients. The present study shows
that the fourth-order compact ADI scheme is stable and efficient
High-order Methods for a Pressure Poisson Equation Reformulation of the Navier-Stokes Equations with Electric Boundary Conditions
Pressure Poisson equation (PPE) reformulations of the incompressible
Navier-Stokes equations (NSE) replace the incompressibility constraint by a
Poisson equation for the pressure and a suitable choice of boundary conditions.
This yields a time-evolution equation for the velocity field only, with the
pressure gradient acting as a nonlocal operator. Thus, numerical methods based
on PPE reformulations, in principle, have no limitations in achieving high
order. In this paper, it is studied to what extent high-order methods for the
NSE can be obtained from a specific PPE reformulation with electric boundary
conditions (EBC). To that end, implicit-explicit (IMEX) time-stepping is used
to decouple the pressure solve from the velocity update, while avoiding a
parabolic time-step restriction; and mixed finite elements are used in space,
to capture the structure imposed by the EBC. Via numerical examples, it is
demonstrated that the methodology can yield at least third order accuracy in
space and time
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