Computational Approaches to the Magnetic Induction Equation in Magnetohydrodynamics (MHD)

Abstract

This study presents a numerical investigation of the magnetic diffusion equation in electrically conducting media using various computational frameworks. The research will model and analyze the time and space dependence of the magnetic field in a current-carrying medium, while investigating the impact of conductive and magnetic properties on diffusion processes. The equation was obtained from Maxwell’s equations combined with Ohm’s law, forming the basis of the parabolic partial differential equation, which is central to magnetohydrodynamics and electrodynamics. To solve this equation for constant parameters, inhomogeneous conditions, and non-linear conductivity in a MATLAB environment, five numerical schemes were used: Forward-Time Central-Space (FTCS), Backward-Time Central-Space (BTCS), the Finite Volume Method (FVM), the Spectral Method (SM), and the Discontinuous Galerkin Method (DGM). The accuracy of the schemes was tested against analytical solutions, using copper properties as an example. The calculation results confirmed the validity of the selected methods, which were used to investigate the dependence of magnetic field penetration and diffusion on electrical conductivity and magnetic permeability. The results showed that higher values of electrical conductivity and magnetic permeability significantly reduce the magnetic field’s penetration and diffusion rate. Furthermore, spatial conductivity gradients and nonlinear behavior introduce distinct asymmetric diffusion profiles. Among the evaluated schemes, the implicit and advanced formulations demonstrated excellent stability and minimal error, strictly adhering to physical benchmarks and relevant Courant-Friedrichs-Lewy (CFL) condition limits. Overall, this study establishes a robust, highly accurate framework for simulating multi-regime magnetic diffusion in conductive media.