Numerical method for the magnetic vector potential in incompressible magnetohydrodynamic flows and the conservation properties of magnetic helicity

Abstract: Analyzing magnetohydrodynamic (MHD) flows requires accurate predictions of the Lorentz force and energy conversion. Total energy, cross-helicity, and magnetic helicity can be used to investigate energy conservation properties in inviscid MHD flows. However, the conservation property of magnetic helicity has not been fully clarified using the magnetic vector potential equation. This study presents a numerical method to simultaneously relax magnetic vector and electric potentials for incompressible MHD flows using a conservative finite difference scheme that discretely conserves total energy. First, it was proven that the transport equations of total energy, cross-helicity, and magnetic helicity can be discretely derived from the equations of momentum, magnetic flux density, and magnetic vector potential, thereby elucidating the conservation properties of these quantities. Subsequently, five models for steady and unsteady problems were analyzed to verify the accuracy and convergence of the proposed numerical method. Additionally, the computational approach involving the magnetic vector and electric potentials was validated. A comparison of the calculated results with exact solutions in the analysis of one- and two-dimensional flow models and Hartmann flow further validated the numerical method. Unsteady analyses of two- and three-dimensional decaying vortices were performed. The ideal periodic inviscid MHD flow exhibited good conservation properties for total energy and cross-helicity. Magnetic helicity was discretely preserved even in three-dimensional flow. Furthermore, in viscous flow, the attenuation trends of total energy, cross-helicity, and magnetic helicity aligned with the exact solution. The numerical method accurately captured the decay trends of energy. Thus, the proposed method can facilitate the investigation of energy conservation and conversion in compressible MHD flows.