Lattice Boltzmann Formulation for Eight-Wave Magnetohydrodynamics

Magnetohydrodynamics couples the Navier–Stokes and Maxwell equations to describes flows in electrically conducting fluids. The divergence of the magnetic field must vanish, but numerical algorithms typically do not preserve this condition exactly. Artifacts can then arise in solutions, such as spurious forces parallel to the magnetic field. These artifacts can be alleviated by using extended sets of Maxwell’s equations that include magnetic charges and currents and hence are invariant under duality rotations that interchange the electric and magnetic fields. The eight-wave formulation supposes that the magnetic current arises from magnetic charges being advected by the fluid. The magnetohydrodynamic equations are then Galilean invariant even when the divergence of the magnetic field is nonzero. The evolution equation for the magnetic field then resembles Jeffery’s equation that describes the orientations of a suspension of axisymmetric particles. The ideal electric field is invariant under the extra terms proportional to the divergence of the magnetic field. This property leads to particularly simple lattice Boltzmann formulations for two of the three variants, with different treatments of the momentum equation, that are constructed and compared. The formulation that implements the Lorentz force directly, rather than via the Maxwell stress, proves more stable in numerical experiments.