Numerical investigation on performance of three solution reconstructions at cell interface in DVM simulation of flows in all Knudsen number regimes

In the conventional discrete velocity method (DVM), the local solution of collisionless Boltzmann equation with a piecewise constant distribution for the distribution function is utilized to reconstruct distribution function at the cell interface and then calculate numerical flux of Boltzmann equation for updating the distribution function at cell center. In this process, a numerical dissipation will be introduced into the solution due to neglecting of the collision effect at the cell interface. This numerical dissipation may deteriorate the solution accuracy of conventional DVM in the continuum flow regime, in which the particle collision happens frequently. To overcome this defect, two improved schemes are first presented in this work, in which the local discrete solution of Boltzmann equation with Shakhov model is adopted to evaluate the distribution function at the cell interface, while the equilibrium state of the local solution is computed by different ways. One of the improved schemes evaluates the equilibrium state exactly by the moments of distribution functions according to the compatibility condition, while the other computes the equilibrium state approximately by a simple average at the cell interface. Since the collision effect is incorporated in evaluation of numerical flux, the improved schemes can provide reasonable solutions in all flow regimes. On the other hand, they introduce some extra computational efforts for determining the collision term at the cell interface as compared with the conventional DVM. To assess the performance of different methods for simulation of flows in all flow regimes, a comprehensive study is then carried out in this work.