Initialization of a Lattice Boltzmann Model with Constrained Runs (Extended Version)

In this article, we perform a numerical stability and convergence analysis of the constrained runs initialization scheme for a lattice Boltzmann model. Gear and Kevrekidis developed this scheme in the context of coarse-grained equation-free computing. Given the macroscopic initial fields, we study the mapping of these variables to the higher-dimensional space of lattice Boltzmann variables. The lattice Boltzmann model considered is the BGK collision model for one-dimensional reaction-diffusion systems. For such systems, we prove that the constrained runs scheme is unconditionally stable and that it converges to an approximation of the slaved state, i.e. the microscopic state consistent with the macroscopic initial condition. This approximation is correct up to first order in the Chapman-Enskog expansion of the lattice Boltzmann model. The asymptotic convergence factor is |1 − ω| with ω the BGK relaxation parameter. The results are illustrated for the one-dimensional FitzHugh-Nagumo reaction-diffusion system. Using the constrained runs initialization in the context of equation-free computing, we also perform a coarse-grained bifurcation analysis of this model. Abstract In this article, we perform a numerical stability and convergence analysis of the constrained runs initialization scheme for a lattice Boltzmann model. Gear and Kevrekidis developed this scheme in the context of coarse-grained equation-free computing. Given the macroscopic initial fields, we study the mapping of these variables to the higher-dimensional space of lattice Boltzmann variables. The lattice Boltzmann model considered is the BGK collision model for one-dimensional reaction-diffusion systems. For such systems, we prove that the constrained runs scheme is unconditionally stable and that it converges to an approximation of the slaved state, i.e. the microscopic state consistent with the macroscopic initial condition. This approximation is correct up to first order in the Chapman-Enskog expansion of the lattice Boltzmann model. The asymptotic convergence factor is |1−ω| with ω the BGK relaxation parameter. The results are illustrated for the one-dimensional FitzHugh-Nagumo reaction-diffusion system. Using the constrained runs initialization in the context of equation-free computing, we also perform a coarse-grained bifurcation analysis of this model.