Vector Lattice Boltzmann Equations: From Magnetohydrodynamics to Active Matter

We present a lattice Boltzmann algorithm for simulating magnetohydrodynamics, and extend it to simulate the Jeffery equation that describes the rotating orientations of axisymmetric particles in a dilute suspension. Both systems involve material vector fields that evolve through the curl of another vector field. Both systems thus require an underlying kinetic formulation using vector fields, in contrast to the scalar fields used in the Boltzmann equation, and in lattice Boltzmann algorithms for hydrodynamics. Simulating Jeffery's equation requires extra gradient terms that cannot be written in conservation form. These gradients are obtained locally at grid points using the non-equilibrium parts of the kinetic vector fields representing the particle orientations, and the kinetic scalar fields representing the suspending fluid. The kinetic formulation is discretised using a Strang splitting between advection to neighbouring grid points and local algebraic operations at grid points.