Convergence of approximate deconvolution models to the mean magnetohydrodynamics equations: Analysis of two models

Abstract

We consider two Large Eddy Simulation (LES) models for the approximation of large scales of the equations of Magnetohydrodynamics (MHD in the sequel). We study two α-models, which are obtained adapting to the MHD the approach by Stolz and Adams with van Cittert approximate deconvolution operators. First, we prove existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations. Then we study the behavior of solutions as the deconvolution parameter goes to infinity. The main result of this paper is the convergence to a solution of the filtered MHD equations. In the final section we study also the problem with filtering acting only on the velocity equation.

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

“…We now give the elements to prove our main result, ie, the existence of an inertial manifold to the system (24)- (27) (which is equivalent to (6)- (8) in the sense of Theorem 4.3 below.…”

“…Here, we consider the approximate deconvolution model, introduced by Adams and Stolz (see also previous studies); by following this scheme, we approximate the filtered bilinear terms as follows: where v and φ play the role of the variables u and θ , respectively, and the filtering operator G α is defined by the Helmholtz filter (see, eg, other studies; see also Bisconti and Catania for an analogous case involving an anisotropic horizontal filter), with and G α :=( I − α 2 Δ) −1 . Here, D N is the deconvolution operator, which is constructed using the Van Cittert algorithm (see, eg, Lewandowski) and is formally defined by …”

“…Here, we consider the approximate deconvolution model, introduced by Adams and Stolz [5][6][7] (see also previous studies 3,[8][9][10] ); by following this scheme, we approximate the filtered bilinear terms as follows:…”

Existence and Convergence of an MHD Approximate Deconvolution Model

“…We briefly present the main ideas of the proof ; for further details and extensions, and for a wider bibliography, see [3].…”

“…For simplicity, we will present here just the first case with double filtering (see [3] for the other case). In order to handle better the approach with two filters, we will use the notation associated to the operators G i .…”