Lifting in hybrid lattice Boltzmann and PDE models

Abstract

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.

Initialization of Lattice Boltzmann Models with the Help of the Numerical Chapman–Enskog Expansion

“…This function takes as input α and β, the vectors of coefficients of expansion (7), and as parameters the fixed density ρ and an integer m, the order of the smoothness condition. It first constructs, with this input, a state (f 1 , f 0 , f −1 ) T with the help of series (7). This state is then used as an initial state to perform a sequence of multiple LBM steps.…”

Numerical Extraction of a Macroscopic PDE and a Lifting Operator from a Lattice Boltzmann Model

“…In [33] the CR-algorithm is combined with Newton's method by performing local updates at the ghost points of the hybrid model to reduce the size of the Jacobian.…”

“…The coupling -that will be discussed in this article -of LBMs with reaction-diffusion PDEs is earlier considered in [29,28,33].…”

Constrained Runs Algorithm as a Lifting Operator for the One-Dimensional in Space Boltzmann Equation with BGK Collision Term

“…Lots of previous work in the literature is based on the initialization of lattice Boltzmann models (LBMs), special discretizations of the Boltzmann equation [19,31]. Lifting operators for LBMs are considered in [22,23,24,26,27,28,30] where a given initial density is mapped to distribution functions. The Chapman-Enskog expansion is built for such model problems in [22].…”

“…The higher order velocity moments become slaved functionals of the density in the LBM context. [27] compares the Chapman-Enskog expansion with the CR algorithm for hybrid model problems with given initial density. [28] constructs the numerical Chapman-Enskog expansion as a lifting operator for these model problems.…”