Simultaneous-approximation-term based boundary discretization for moment equations of rarefied gas dynamics

Abstract

Previous works have developed boundary conditions that lead to the L2-boundedness of solutions to the linearised moment equations. Here we present a spatial discretization that preserves the L2-stability by recovering integration-by-parts over the discretized domain and by imposing boundary conditions using a simultaneous-approximation-term (SAT). We develop three different forms of the SAT using: (i) characteristic splitting of moment equation's boundary conditions; (ii) decoupling of moments in moment equations; and (iii) characteristic splitting of Boltzmann equation's boundary conditions. We discuss how the first two forms differ in terms of their usage and implementation. We show that the third form is equivalent to using an upwind kinetic numerical flux along the boundary, and we argue that even though it provides stability, it prescribes the incorrect number of boundary conditions. Using benchmark problems, we compare the accuracy of moment solutions computed using different SATs. Our numerical experiments also provide new insights into the convergence of moment approximations to the Boltzmann equation's solution.