  We present a multi-scale lattice Boltzmann scheme, which adaptively refines
particles' velocity space. Different velocity sets, i.e., higher- and
lower-order lattices, are consistently and efficiently coupled, allowing us to
use the higher-order lattice only when and where needed. This includes regions
of either high Mach number or high Knudsen number. The coupling procedure of
different lattices consists of either projection of the moments of the
higher-order lattice onto the lower-order lattice or lifting of the lower-order
lattice to the higher-order velocity space. Both lifting and projection are
local operations, which enable a flexible adaptive velocity set. The proposed
scheme can be formulated both in a static and an optimal, co-moving reference
frame, in the spirit of the recently introduced Particles on Demand method. The
multi-scale scheme is first validated through a convected athermal vortex and
also studied in a jet flow setup. The performance of the proposed scheme is
further investigated through the shock structure problem and a high Knudsen
Couette flow, typical examples of highly non-equilibrium flows in which the
order of the velocity set plays a decisive role. The results demonstrate that
the proposed multi-scale scheme can operate accurately, with flexibility in
terms of the underlying models and with reduced computational requirements.
