Moment-Based Boundary Conditions for Lattice Boltzmann Magnetohydrodynamics

We present a moment-based approach for implementing boundary conditions in a lattice Boltzmann formulation of magnetohydrodynamics. Hydrodynamic quantities are represented using a discrete set of distribution functions that evolve according to a cut-down form of Boltzmann’s equation from continuum kinetic theory. Electromagnetic quantities are represented using a set of vector-valued distribution functions. The nonlinear partial differential equations of magnetohydrodynamics are thus replaced by two constant-coefficient hyperbolic systems in which all nonlinearities are confined to algebraic source terms. Further discretising these systems in space and time leads to efficient and readily parallelisable algorithms. However, the widely used bounce-back boundary conditions place no-slip boundaries approximately half-way between grid points, with the precise position being a function of the viscosity and resistivity. Like most lattice Boltzmann boundary conditions, bounce-back is inspired by a discrete analogue of the diffuse and specular reflecting boundary conditions from continuum kinetic theory. Our alternative approach using moments imposes no-slip boundary conditions precisely at grid points, as demonstrated using simulations of Hartmann flow between two parallel planes.