  We propose a new spectral Lagrangian based deterministic solver for the
non-linear Boltzmann Transport Equation for Variable Hard Potential (VHP)
collision kernels with conservative or non-conservative binary interactions.
The method is based on symmetries of the Fourier transform of the collision
integral, where the complexity in its computing is reduced to a separate
integral over the unit sphere $S^2$. In addition, the conservation of moments
is enforced by Lagrangian constraints. The resulting scheme, implemented in
free space is very versatile and adjusts in a very simple manner, to several
cases that involve energy dissipation due to local micro-reversibility
(inelastic interactions) or elastic model of slowing down process. Our
simulations are benchmarked with the available exact self-similar solutions,
exact moment equations and analytical estimates for homogeneous Boltzmann
equation for both elastic and inelastic VHP interactions. Benchmarking of the
simulations involves the selection of a time self-similar rescaling of the
numerical distribution function which is performed using the continuous
spectrum of the equation for Maxwell molecules as studied first in
\cite{asympGranular} and generalized to a wide range of related models in
\cite{BCG}. The method also produces accurate results in the case of inelastic
diffusive Boltzmann equations for hard-spheres (inelastic collisions under
thermal bath), where overpopulated non-Gaussian exponential tails have been
conjectured in computations by stochastic methods in \cite{vanNoi-Ernst,
ernst-brito, MSS, GRW04} and rigourously proven in \cite{diffGran} and
\cite{highEnergyTails}.
