Hyperbolic Divergence Cleaning in Lattice Boltzmann Magnetohydrodynamics (Communications in Computational Physics, 2023)
Magnetohydrodynamics couples the Navier–Stokes and Maxwell’s equations to describe the flow of electrically conducting fluids in magnetic fields. Maxwell’sequations require the divergence of the magnetic field to vanish, but this condition istypically not preserved exactly by numerical algorithms. Solutions can develop artifacts because structural properties of the magnetohydrodynamic equations then fail tohold. Magnetohydrodynamics with hyperbolic divergence cleaning permits a nonzerodivergence that evolves under a telegraph equation, designed to both damp the divergence, and propagate it away from any sources, such as poorly resolved regions withlarge spatial gradients, without significantly increasing the computational cost. Weshow that existing lattice Boltzmann algorithms for magnetohydrodynamics alreadyincorporate hyperbolic divergence cleaning, though they typically use parameter values for which it reduces to parabolic divergence cleaning under a slowly-varying approximation. We recover hyperbolic divergence cleaning by adjusting the relaxationrate for the trace of the tensor that represents the electric field, and absorb the contribution from the symmetric-traceless part of this tensor using a change of variables.Numerical experiments confirm that the qualitative behaviour changes from parabolicto hyperbolic when the relaxation time for the trace of the electric field tensor is increased.
Keywords: Magnetohydrodynamics; divergence cleaning; telegraph equations; matrix collision operators; linear viscoelasticity.