README.md

2026GJZM

Codex/ChatGPT (July 2026)

Summary

The authors construct a linear, second-order scheme for three-dimensional incompressible Hall magnetohydrodynamics on $\mathbb{R}^3$, avoiding artificial domain truncation. Mapped Gegenbauer functions provide global spatial approximation, while a non-zero auxiliary variable linearizes nonlinear advection, Lorentz-force, and Hall terms. A Lagrange multiplier enforces the magnetic divergence constraint weakly, and a projection method decouples velocity and pressure through Stokes solves. Time-stepping-varying Crank--Nicolson discretization supports adaptive steps and preserves an unconditional discrete energy-dissipation law. Manufactured-solution tests show exponential spatial convergence and second-order temporal accuracy; adaptive stepping tracks rapid early energy decay while reducing CPU time. Energy remains monotone for time steps as large as one and across viscosity, diffusivity, and Hall-parameter tests. Simulations of magnetic $X$- and $O$-points further reproduce Hall-driven current-layer intensification, with current magnitude increasing as the Hall parameter grows.

Contributions

1. Formulated a Gegenbauer--Galerkin discretization for the full unbounded three-dimensional domain.
2. Combined an auxiliary variable, magnetic Lagrange multiplier, and projection method into a linear algorithm.
3. Proved weak magnetic divergence preservation, second-order accuracy, and unconditional energy stability.
4. Demonstrated exponential spatial convergence and effective adaptive time stepping numerically.
5. Recovered Hall-induced current-density intensification during magnetic-null evolution.