Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann
equation employ a point measure approximation to the distribution
function, as simulated particles may possess only a single velocity.
This representation limits the method to converge only weakly to
the solution of the Boltzmann equation. Utilizing kernel density
estimation we have developed a stochastic Boltzmann solver which
possesses strong convergence for bounded and $L^\infty$ solutions
of the Boltzmann equation. This is facilitated by distributing
the velocity of each simulated particle instead of using the
point measure approximation inherent to DSMC. We propose that the
development of a distributional method which incorporates distributed
velocities in collision selection and modeling should improve convergence
and potentially result in a substantial reduction of the variance in
comparison to DSMC methods. Toward this end, we also report initial
findings of modeling collisions distributionally using the
Bhatnagar-Gross-Krook collision operator.