Quadrature-Based Lattice Boltzmann Models for Rarefied Gas Flow

At the microscale, experiments show that in the vicinity of a moving boundary, rarefied fluid flows exhibit a velocity slip (i.e., the fluid and wall velocities are not equal), as well as a temperature jump (i.e., the fluid temperature is not equal to the wall temperature). Such effects can be captured within the framework of the Boltzmann equation by employing kinetic boundary conditions (i.e., diffuse reflection). In our contribution, we present the systematic construction of lattice Boltzmann (LB) models based on Gauss quadratures, which allows one to simulate gas flows between diffuse reflecting boundaries from the hydrodynamic limit to the ballistic regime. Key to the success of this approach is the use of half-range Gauss–Hermite quadratures, which are essential in order to capture the discontinuity of the distribution function induced by the boundaries. Since the resulting models are off-lattice, an overview of the appropriate numerical schemes for these models is also provided.