In this paper, the previously described monolithic approach  for the stationary discrete Boltzmann equation is extended to time-dependent problems. In general, both collision and advection operators are discretized on nonuniform grids as opposed to the standard Lattice Boltzmann method. Implicit time-stepping schemes are applied for an accurate and robust numerical treatment of the nonstationary flow problems. The resulting coupled system of equations is treated using special numerical methods for PDE’s. As in the steady case, we apply a full Newton method for the nonlinear problems, but we also discuss possible variants of semi-implicit schemes all of which lead to nonsymmetric linear systems. The preconditioning of the used Krylov-space methods, resp., the construction of corresponding smoothing operators in the applied multigrid approaches is closely connected to the underlying short characteristic-upwinding discretization, yielding the exact inverse of the transport operators even for unstructured meshes due to a special numbering technique. Numerical results are given for the proposed solvers analysing the efficiency depending on the Mach number, time step and mesh size, while accuracy and stability of the complete space-time discretization are demonstrated for prototypical flow configurations at various timesteps.
© 2014 by Walter de Gruyter Berlin/Boston