Abstract <p>An algorithm for calculating the Boltzmann collision integral for a small impurity in a background gas has been developed. The method is able to calculate the evolution of the impurity distribution function on a compact local velocity grid that can be significantly smaller than the domain of the background distribution. The algorithm is based on preliminary computation of the transition frequency matrix from one grid cell to another. Calculation of the frequency matrix is reduced to integration over a plane in velocity space determined by the energy and momentum conservation laws. A modification of the method that strictly preserves the number of particles in the computational domain is presented. The method is extended to the case of arbitrary cross sections and inelastic collisions. Tests for relaxation to a Maxwellian distribution, scattering by flow, and beam deceleration confirm the accuracy and applicability of the method to problems of energy and momentum transfer in mixtures with widely varying temperatures.</p>