  Simulating plasmas in the Hall-MagnetoHydroDynamics (Hall-MHD) regime
represents a valuable {approach for the investigation of} complex non-linear
dynamics developing in astrophysical {frameworks} and {fusion machines}. Taking
into account the Hall electric field is {computationally very challenging as}
it involves {the integration of} an additional term, proportional to $\bNabla
\times ((\bNabla\times\mathbf{B})\times \mathbf{B})$ in the Faraday's induction
{law}. {The latter feeds back on} the magnetic field $\mathbf{B}$ at small
scales (between the ion and electron inertial scales), {requiring} very high
resolution{s} in both space and time {in order to properly describe its
dynamics.} The computational {advantage provided by the} kinetic Lattice
Boltzmann (LB) approach is {exploited here to develop a new} code, the
\textbf{\textsc{F}}ast \textbf{\textsc{L}}attice-Boltzmann
\textbf{\textsc{A}}lgorithm for \textbf{\textsc{M}}hd
\textbf{\textsc{E}}xperiments (\textsc{flame}). The \textsc{flame} code
integrates the plasma dynamics in lattice units coupling two kinetic schemes,
one for the fluid protons (including the Lorentz force), the other to solve the
induction equation describing the evolution of the magnetic field. Here, the
newly developed algorithm is tested against an analytical wave-solution of the
dissipative Hall-MHD equations, pointing out its stability and second-order
convergence, over a wide range of the control parameters. Spectral properties
of the simulated plasma are finally compared with those obtained from numerical
solutions from the well-established pseudo-spectral code \textsc{ghost}.
Furthermore, the LB simulations we present, varying the Hall parameter,
highlightthe transition from the MHD to the Hall-MHD regime, in excellent
agreement with the magnetic field spectra measured in the solar wind.
