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.