A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfvénic alignment in periodic domains
Abstract
Cross helicity, together with total energy, determines the Elsässer energies and thus quantifies departures from Alfvénic alignment in incompressible MHD. Building on the dual-field method in [Mao and Xi, \emph{J. Comput. Phys.}, 2025], we construct a linear finite element method for periodic domains that reproduces the cross-helicity balance. We connect the discrete energy and cross-helicity conservation laws to Alfvénic alignment through a nonnegative Elsässer-energy representation, establishing preservation of the corresponding exact relation when either Elsässer energy is initially zero in the unforced ideal case. We reassign the two discrete approximations of the magnetic field to the integer and half-integer time levels and redesign the Lorentz and induction couplings within the Mao-Xi formulation. The resulting scheme uses linear staggered updates and retains the energy balances, strong and weak divergence-free properties, and magnetic- and fluid-helicity balances. Numerical experiments illustrate the role of the cross-helicity balance in preserving an aligned state in the ideal case and in alignment dynamics under weak dissipation.
Related
- A robust finite element method for incompressible magnetohydrodynamics using Whitney forms
- A dual-field structure-preserving mixed finite element discretization for incompressible Hall MHD equations
- Mini mixed finite element method for nearly incompressible linear elasticity problems
- A Quantum Spectral Solver for Periodic Incompressible Stokes Flow
- A Modified Brinkman-Type Fictitious-Domain Finite Element Method for Incompressible Flow with Rigid Obstacles
BibTeX
@misc{li2026linear,
title = {{A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfvénic alignment in periodic domains}},
author = {Shusheng Li and Siyuan Tong and Qilong Zhai and Qian Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32476},
eprint = {2610.32476},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32476},
}
RIS
TY - UNPB
TI - A linear dual-field finite element method for incompressible MHD, part I: cross-helicity and Alfvénic alignment in periodic domains
AU - Shusheng Li
AU - Siyuan Tong
AU - Qilong Zhai
AU - Qian Zhang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32476
UR - https://arxiv.org/abs/2610.32476
AB - Cross helicity, together with total energy, determines the Elsässer energies and thus quantifies departures from Alfvénic alignment in incompressible MHD. Building on the dual-field method in [Mao and Xi, \emph{J. Comput. Phys.}, 2025], we construct a linear finite element method for periodic domains that reproduces the cross-helicity balance. We connect the discrete energy and cross-helicity conservation laws to Alfvénic alignment through a nonnegative Elsässer-energy representation, establishing preservation of the corresponding exact relation when either Elsässer energy is initially zero in the unforced ideal case. We reassign the two discrete approximations of the magnetic field to the integer and half-integer time levels and redesign the Lorentz and induction couplings within the Mao-Xi formulation. The resulting scheme uses linear staggered updates and retains the energy balances, strong and weak divergence-free properties, and magnetic- and fluid-helicity balances. Numerical experiments illustrate the role of the cross-helicity balance in preserving an aligned state in the ideal case and in alignment dynamics under weak dissipation.
ER -