High-Order Curvilinear Finite Element Magneto-Hydrodynamics I: A Conservative Lagrangian Scheme

TL;DR

This paper introduces a high-order conservative Lagrangian finite element method for magneto-hydrodynamics that preserves key physical invariants and divergence-free magnetic fields, enabling accurate simulations of complex plasma phenomena.

Contribution

It presents the first high-order multi-dimensional conservative Lagrangian scheme based on curvilinear finite elements for magneto-hydrodynamics, ensuring exact invariance and divergence-free magnetic fields.

Findings

Exact preservation of divergence-free magnetic field.

High-order convergence achieved in ideal MHD problems.

Effective handling of resistive magnetic diffusion with implicit schemes.

Abstract

Magneto-hydrodynamics is one of the foremost models in plasma physics with applications in inertial confinement fusion, astrophysics and elsewhere. Advanced numerical methods are needed to get an insight into the complex physical phenomena. The classical Lagrangian methods are typically limited to the low orders of convergence and suffer from violation of the divergence-free condition for magnetic field or conservation of the invariants. This paper is the first part of a new series about high-order non-ideal magneto-hydrodynamics, where a multi-dimensional conservative Lagrangian method based on curvilinear finite elements is presented. The condition on zero divergence of magnetic field and conservation of mass, momentum, magnetic flux and the total energy are satisfied exactly. The curvilinear elements prevent entangling of the computational mesh and its imprinting into the solution. A high-order conservative time integration is applied, where an arbitrary order of convergence is attained for problems of ideal magneto-hydrodynamics. The resistive magnetic field diffusion is solved by an implicit scheme. Description of the method is given and multiple test problems demonstrating properties of the scheme are performed. The construction of the method and possible future directions of development are discussed.