  A lattice Boltzmann scheme is presented which recovers the dynamics of
nematic and chiral liquid crystals; the method essentially gives solutions to
the Qian-Sheng equations for the evolution of the velocity and tensor
order-parameter fields. The resulting algorithm is able to include five
independent Leslie viscosities, a Landau-deGennes free energy which introduces
three or more elastic constants, a temperature dependent order parameter,
surface anchoring and viscosity coefficients, flexo-electric and order
electricity and chirality. When combined with a solver for the Maxwell
equations associated with the electric field, the algorithm is able to provide
a full 'device solver' for a liquid crystal display. Coupled lattice Boltzmann
schemes are used to capture the evolution of the fast momentum and slow
director motions in a computationally efficient way. The method is shown to
give results in close agreement with analytical results for a number of
validating examples. The use of the method is illustrated through the
simulation of the motion of defects in a zenithal bistable liquid crystal
device.
