  The lattice Boltzmann method (LBM) is a numerical approach to tackle problems
described by a Boltzmann type-equation, where time, space, and velocities are
discretized to describe scattering and advection. Even though the LBM executes
advection along a lattice direction without numerical error, its usage in the
high Knudsen number regime (ballistic) has been hindered by the ray effect
problem (for dimensions greater than 1D). This problem has its origin in the
low number of available propagation directions on standard LBM lattices. Here,
to overcome this limitation, we propose the worm-lattice Boltzmann method
(worm-LBM), which allows a high number of lattice directions by alternating in
time the basic directions described within the next neighbor schemes.
Additionally, to overcome the velocity anisotropy issue, which otherwise
clearly manifests itself in the ballistic regime (e.g. the $\sqrt 2$ higher
grid velocity of the D2Q8 scheme along the diagonal direction compared to the
axial one), the time-adaptive scheme (TAS) is proposed. The TAS method makes
use of pausing advection on the grid, allowing to impose not only isotropic
propagation but also arbitrary direction-dependent grid velocity. Last but not
least, we propose a grid-mean free path (grid-MFP) correction to correctly
handle the aforementioned velocity issue in the diffusive limit, without
affecting the ballistic one. We provide a detailed description of the TAS
method and the worm-LBM algorithm, and verify their numerical accuracy by using
several transient diffusive-ballistic phonon transport cases, including
different initial and boundary conditions. Overall, the new, very accurate, and
efficient worm-LBM algorithm, free of numerical smearing and false scattering,
has the potential to be at the forefront of the numerical solvers to tackle the
advective part of different equations in a wide field of applications.
