Improvement Of A Discrete Velocity Boltzmann Equation Solver With Arbitrary Post-Collision Velocities

We present a discrete velocity scheme which solves the Boltzmann equation and show numerical results for homogeneous relaxation problems. Although direct simulation of the Boltzmann equation can be efficient for transient problems, computational costs have restricted its use. A velocity interpolation algorithm enables us to select post-collision velocity pairs not restricted to those that lie precisely on the grid. This allows efficient evaluation of the replenishing part of the collision integral with reasonable accuracy. In previous work [1] the scheme was demonstrated with the depleting terms evaluated exactly, which made the method of O(N(2)) where N is the number of grid points in the velocity space. In order to reduce the computational cost, we have developed an acceptance-rejection scheme to enable more efficient evaluation of the depleting term. We show that the total collision integral can be evaluated accurately in combination with the mapping scheme for the replenishing term. To improve our scheme, we study the error and computational time associated with the number of depleting and replenishing points. We predict the correct relaxation rate for the Bobylev-Krook-Wu distribution and obtain exact conservation of mass, momentum, and energy. Comparisons between computed and reference solutions are shown as well, demonstrating the correct relaxation rate and dependence of error on parameters in the computational scheme.