Efficient simulation techniques of the Lattice Boltzmann equation on unstructured meshes

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 differencelike 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.