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

Abstract

Lifting operators play an important role in starting a lattice Boltzmann model from a given initial density. The density, a macroscopic variable, needs to be mapped to the distribution functions, mesoscopic variables, of the lattice Boltzmann model. Several methods proposed as lifting operators have been tested and discussed in the literature. The most famous methods are an analytically found lifting operator, like the Chapman-Enskog expansion, and a numerical method, like the Constrained Runs algorithm, to arrive at an implicit expression for the unknown distribution functions with the help of the density. This paper proposes a lifting operator that alleviates several drawbacks of these existing methods. In particular, we focus on the computational expense and the analytical work that needs to be done. The proposed lifting operator, a numerical Chapman-Enskog expansion, obtains the coefficients of the Chapman-Enskog expansion numerically. Another important feature of the use of lifting operators is found in hybrid models. There the lattice Boltzmann model is spatially coupled with a model based on a more macroscopic description, for example an advection-diffusionreaction equation. In one part of the domain, the lattice Boltzmann model is used, while in another part, the more macroscopic model. Such a hybrid coupling results in missing data at the interfaces between the different models. A lifting operator is then an important tool since the lattice Boltzmann model is typically described by more variables than a model based on a macroscopic partial differential equation.

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

“…Instead of using CR to find for each grid point the missing moments φ and ξ, the numerical Chapman-Enskog expansion uses CR to find the unknown coefficients of the expansion [8]. The major advantage is that it leads to much smaller systems since it finds the expansion coefficients that satisfy the smoothness condition rather than the full state of the velocity moments.…”

“…once the distribution functions are close to the slow manifold since on the slow manifold the distribution functions are parametrized by the density [8]. x j and x k represent grid points of the spatial domain.…”

“…Lifting operators for LBMs are considered in [5,6,7,8,9,10] where density is mapped to distribution functions. The Chapman-Enskog expansion is built for such model problems in [5].…”

“…The Chapman-Enskog expansion needs to be constructed analytically while the CR algorithm is computationally expensive, especially for higher dimensional problems. Because of these drawbacks, [8] constructs the numerical Chapman-Enskog expansion as a lifting operator for these model problems. This numerical Chapman-Enskog expansion is based on a combination of the Chapman-Enskog expansion and the CR algorithm.…”

“…The goal of this paper is to generalize it to models where both density and momentum are conserved. We borrow the concept of the numerical Chapman-Enskog expansion constructed in [8] and use it to map both density and momentum to distribution functions of the LBM. The proposed algorithm is an extension that shows that the numerical Chapman-Enskog expansion can be generalized for the considered model problems.…”