Newton–Krylov solvers for the equation-free computation of coarse traveling waves

For many complex dynamical systems, a separation of scales prevails between the (fine-scale) level of description of the available model, and the (coarse) 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 fine-scale simulations, one can build a coarse time-stepper to approximate a time-stepper for the unavailable coarse model. Here, we use this coarse time-stepper to estimate matrix–vector products in a Jacobian-free Newton–GMRES method. The GMRES convergence is accelerated with a preconditioner that is derived from an approximate coarse equation. We examine the numerical properties of the approach with the computation of coarse traveling wave solutions of two lattice Boltzmann models for planar streamer fronts.