  Several applications exist in which lattice Boltzmann methods (LBM) are used
to compute stationary states of fluid motions, particularly those driven or
modulated by external forces. Standard LBM, being explicit time-marching in
nature, requires a long time to attain steady state convergence, particularly
at low Mach numbers due to the disparity in characteristic speeds of
propagation of different quantities. In this paper, we present a preconditioned
generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate
steady state convergence to flows driven by external forces. The use of
multiple relaxation times in the GLBE allows enhancement of the numerical
stability. Particular focus is given in preconditioning external forces, which
can be spatially and temporally dependent. In particular, correct forms of
moment-projections of source/forcing terms are derived such that they recover
preconditioned Navier-Stokes equations with non-uniform external forces. As an
illustration, we solve an extended system with a preconditioned lattice kinetic
equation for magnetic induction field at low magnetic Prandtl numbers, which
imposes Lorentz forces on the flow of conducting fluids. Computational studies,
particularly in three-dimensions, for canonical problems show that the number
of time steps needed to reach steady state is reduced by orders of magnitude
with preconditioning. In addition, the preconditioning approach resulted in
significantly improved stability characteristics when compared with the
corresponding single relaxation time formulation.
