In this paper, we apply special techniques from Numerics for PDE’s to the Lattice Boltzmann
equation. In [5] the concept of the generalized mean intensity has been proposed for radiative
transfer equations. Here, we adapt this concept to the LBE, treating it as an analogous integro
differential equation with constant characteristics. Thus, we combine an efficient finite difference-
like discretization based on short-characteristic upwinding techniques on unstructured, locally
adapted grids with Krylow-space methods (Bi-CGSTAB, GMRES, etc.). The implicit treatment
of the LBE leads to nonlinearities which are efficiently solved with the Newton method even
for a direct, stationary solution of the LBE. With our direct preconditioning by the transport
part we obtain an efficient linear solver for transport dominated configurations, while collision
dominated cases are treated with a block-Jacobian preconditioning. Due to our new generalized
equilibrium formulation (GEF) we can combine the advantages of both preconditioners, i.e.
independence of the number of unknowns for convection-dominated cases with robustness against
stiff configurations. We further improve the GEF approach by using multigrid algorithms to
obtain very good convergence rates for a wide range of problem parameters, and demonstrate
the results via various benchmark problems.