  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-diffusion-reaction 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.
