D. M. Bond

Solving the discrete S-model kinetic equations with arbitrary order polynomial approximations

A numerical method for solving the model Boltzmann equation using arbitrary order polynomials is presented. The S-model is solved by a discrete ordinate method with velocity space discretized with a truncated Hermite polynomial expansion. Physical space is discretized according to the Conservative Flux Approximation scheme with extension to allow non-uniform grid spacing. This approach, which utilizes Legendre polynomials, allows the spatial representation and flux calculation to be of arbitrary order. High order boundary conditions are implemented. Various results are shown to demonstrate the utility and limitations of the method with comparison to solutions of the Euler and Navier-Stokes equations and from the Direct Simulation Monte Carlo and Unified Gas Kinetic schemes. The effect of both the velocity space discretization and Knudsen number on the convergence properties of the scheme are also investigated.

Comparison of discrete BGK-Shakhov system with DSMC

Results obtained by numerically solving the discrete Boltzmann equation, using the Bhattnagar-Gross-Krook (BGK) relaxation approximation with Shakhov target distribution, are compared to those obtained by the Direct Simulation Monte-Carlo (DSMC) method. The relaxation dynamics of the BGK-Shakhov system are also compared to DSMC results. The method is of arbitrary order with velocity and physical space discretized according to truncated series of Hermite and Legendre polynomials respectively. The polynomial based physical space discretization is shown to allow the implementation of the relaxation operator in a continuous-in-space manner. Results show a high degree of similarity with DSMC solutions with little numerical dissipation and no statistical scatter.