On the convergence of a time discretization scheme for the Navier-Stokes equations

Abstract

Abstract.A linearized version of the implicit Euler scheme is considered for the approximation of the solutions to the Navier-Stokes equations in a two-dimensional domain. The rate of convergence in the if1-norm is established.

Asymptotic behavior and time discretization analysis for the non-stationary Navier-Stokes problem

“…Some similar results were obtained by Geveci [2] for the linear Euler scheme and by Shen [11] and He & Li [5] for the nonlinear Galerkin scheme.…”

Approximating stationary statistical properties

“…(11) Can we further develop this theory and combine it with fluctuation dissipation theory to study climate change in climate models? (12) Many physical models involve randomness. How do we generalize the current theory to infinite dimensional dissipative random dynamical systems?…”

“…Even under the ergodicity assumption, it is not at all clear that classical numerical schemes which provide accurate approximation on finite time interval will remain meaningful for stationary statistical properties (long time properties) since small error will be amplified and accumulated over long time except in the case that the underlying dynamics is asymptotically stable (see [12,14,19]) where statistical approach is not necessary since there is no chaos. Indeed, let S k be the solution operator of a one-step scheme with time step k = ∆t and assume that the scheme is of order m so that the following type of error estimate holds: dist H (S(nk)u, S n k u) ≤ C exp(αnk)k m where C > 0 and α are constants, which would induce on a time interval [0, T ], an a priori error bound on the long time average of the order of k m exp(αT )−exp(αk) exp(αk)−1 which diverges for positive α as T approaches infinity.…”

“…On the other hand, there are few results on the convergence of stationary statistical properties of numerical schemes for chaotic PDEs (see [4,5,41]), although there have been a lot of work on temporal/spatial approximation of dissipative dynamical systems, such as the two dimensional incompressible Navier-Stokes system and the one-dimensional Kuramoto-Sivashinsky equation (see [8,9,12,15,17,28,29,34] among others). These authors were mostly interested in the long time stability of the scheme in the sense of deriving uniform in time bounds on the scheme (sometimes bound in the phase space H only which is not sufficient for uniform dissipativity, although it may be sufficient for the convergence of the global attractors), and approximation of various invariant sets (such as steady states, time periodic orbits, global attractors, inertial manifolds, etc., see [15,16,28,[31][32][33] and the references therein).…”

On the time discretization for the globally modified three dimensional Navier–Stokes equations

“…Numerical schemes that inherit properties of the continuous model are now well documented in the literature (see [4][5][6][7][8][9][10][11][12][13][14][15], among others). In that regard, two dimensional Navier-Stokes equations have been studied in [8,4,6,9,10], while numerical schemes that replicate the a priori estimate of the H 2 norm of the velocity for MHD have been constructed in [5]. Ning Ju [7], has proved the existence of a global attractor in V by using the method presented in [16] [Chap 3], which consists of obtaining some a priori estimate in more regular function spaces and using compact embedding of the Sobolev spaces.…”