Computing and Visualization in Science

, Volume 13, Issue 3, pp 129–143 | Cite as

Efficient monolithic simulation techniques for the stationary Lattice Boltzmann equation on general meshes

  • T. Hübner
  • S. Turek
Regular Article


In this paper, we present special discretization and solution techniques for the numerical simulation of the Lattice Boltzmann equation (LBE). In Hübner and Turek (Computing, 81:281–296, 2007), the concept of the generalized mean intensity had been proposed for radiative transfer equations which we adapt here to the LBE, treating it as an analogous (semi-discretized) 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 fast iterative solvers. The fully implicit treatment of the LBE leads to nonlinear systems which can be efficiently solved with the Newton method, even for a direct solution of the stationary LBE. With special exact preconditioning by the transport part due to the short-characteristic upwinding, we obtain an efficient linear solver for transport dominated configurations (macroscopic Stokes regime), while collision dominated cases (Navier-Stokes regime for larger Re numbers) are treated with a special block-diagonal preconditioning. Due to the 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 for stiff configurations. We further improve the GEF approach by using hierarchical multigrid algorithms to obtain grid-independent convergence rates for a wide range of problem parameters, and provide representative results for various benchmark problems. Finally, we present quantitative comparisons between a highly optimized CFD-solver based on the Navier-Stokes equation (FeatFlow) and our new LBE solver (FeatLBE).


Lattice Boltzmann equation Integro-differential equations Constant characteristics Upwind discretization Lower triangular matrices Krylov-methods Multigrid Monolithic solver 

AMS Subject Classification

35A25 65M06 76D05 76P05 


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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute for Applied MathematicsTU DortmundDortmundGermany

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