LSFEM implementation of MHD numerical solver

TL;DR

This paper introduces a new MHD solver based on Least-Squares Finite Element Method (LSFEM), demonstrating its effectiveness for complex, multi-scale astrophysical plasma problems with initial tests on ideal and resistive MHD cases.

Contribution

It presents the first implementation of an LSFEM-based MHD solver, detailing the formulation, transformation, and practical aspects of the method.

Findings

The LSFEM solver successfully handles ideal and resistive MHD problems.

The method shows promise for multi-scale astrophysical plasma simulations.

Initial tests confirm the usability of the LSFEM approach for MHD equations.

Abstract

Many problems in physics are inherently of multi-scale nature. The issues of MHD turbulence or magnetic reconnection, namely in the hot and sparse, almost collision-less astrophysical plasmas, can stand as clear examples. The Finite Element Method (FEM) with adaptive gridding appears to be the appropriate numerical implementation for handling the broad range of scales contained in such high Lundquist-number MHD problems. In spite the FEM is now routinely used in engineering practice in solid-state and fluid dynamics, its usage for MHD simulations has recently only begun and only few implementations exist so far. In this paper we present our MHD solver based on the Least-Square FEM (LSFEM) formulation. We describe the transformation of the MHD equations into form required for finding the LSFEM functional and some practical issues in implementation of the method. The algorithm was tested on selected problems of ideal (non-resistive) and resistive MHD. The tests show the usability of LSFEM for solving MHD equations.