A nonstationary Boltzmann particle-mesh method for neutral hydrogen originating from either the interstellar medium, the heliosheath, or the solar wind has been developed to simulate the detailed interaction of the solar wind with the local interstellar medium (LISM). Such a Boltzmann description for the hydrogen serves to capture the complicated neutral distribution. The neutral component is coupled to a hydrodynamic plasma through charge exchange. The distribution of the ionized component was taken from a two-shock model [Zank et al., 1996b>. This approach is used (1) to elucidate the nature of the solar wind-LISM interaction more carefully than has been done previously in a multifluid approach [Zank et al., 1996b> and (2) to identify precisely the role played by the three identifiably distinct neutral populations in determining the global structure and dynamics of the heliosphere. The differences in the global neutral H distribution that exist between Boltzmann and multifluid gasdynamic models are discussed briefly. ¿ 1998 American Geophysical Union