Convergence of approximate deconvolution models to the mean Magnetohydrodynamics Equations: Analysis of two models

TL;DR

This paper analyzes two Large Eddy Simulation models for Magnetohydrodynamics, proving their solutions converge to the filtered MHD equations as the deconvolution parameter increases, enhancing understanding of LES modeling in MHD.

Contribution

It introduces and rigorously analyzes two new $ ext{α}$-models for MHD LES, demonstrating convergence to the filtered equations and extending previous deconvolution approaches.

Findings

Existence and uniqueness of regular weak solutions for the models.

Solutions converge to the filtered MHD equations as deconvolution parameter approaches infinity.

Analysis includes models with filtering on both equations and only on the velocity equation.

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 $\alpha$-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.