Coarse-grained computation of traveling waves of lattice Boltzmann models with Newton–Krylov methods

For many complex dynamical systems, a separation of scales prevails between the (microscopic) level of description of the available model, and the (macroscopic) level at which one would like to observe and analyze the system. For this type of problems, an “equation-free” framework has recently been proposed. Using appropriately initialized microscopic simulations, one can build a coarsegrained time-stepper to approximate a time-stepper for the unavailable macroscopic model. Here, we show how one can use this coarsegrained time-stepper to compute coarse-grained traveling wave solutions of a lattice Boltzmann model. In a moving frame, emulated by performing a shift-back operation after the coarse-grained timestep, the traveling wave appears as a steady state, which is computed using an iterative method, such as Newton–GMRES. To accelerate convergence of the GMRES procedure, a macroscopic model-based preconditioner is used, which is derived from the lattice Boltzmann model using a Chapman–Enskog expansion. We illustrate the approach on a lattice Boltzmann model for the Fisher equation and on a model for ionization waves.