Lifting in hybrid lattice Boltzmann and PDE models

Mathematical models based on kinetic equations are ubiquitous in the modeling of granular media, population dynamics of biological colonies, chemical reactions and many other scientific problems. These individual-based models are computationally very expensive because the evolution takes place in the phase space. Hybrid simulations can bring down this computational cost by replacing locally in the domain — in the regions where it is justified — the kinetic model with a more macroscopic description. This splits the computational domain into subdomains. The question is how to couple these models in a mathematically correct way with a lifting operator that maps the variables of the macroscopic partial differential equation to those of the kinetic model. Indeed, a kinetic model has typically more variables than a model based on a macroscopic partial differential equation and at each interface we need the missing data. In this contribution we report on different lifting operators for a hybrid simulation that combines a lattice Boltzmann model — a special discretization of the Boltzmann equation — with a diffusion partial differential equation. We focus on the numerical comparison of various lifting strategies.