A true-direction reconstruction of the quiet direct simulation method for inviscid gas flows

In this paper, a true-direction flux reconstruction of the second-order quiet direct simulation (QDS-2N) Smith et al. (2009) [3] as an equivalent Euler equation solver, called QDS-N², is proposed. Because of the true-directional nature of QDS, where volume-to-volume (true-direction) fluxes are computed, as opposed to fluxes at cell interfaces as employed by traditional finite volume schemes, a volumetric reconstruction is required to reach a totally true-direction scheme. The conserved quantities are permitted to vary (according to a polynomial expression) across all simulated dimensions. Prior to the flux computation, QDS particles are introduced using properties based on weighted moments taken over the polynomial reconstruction of the conserved quantity fields. The resulting flux expressions are shown to exactly reproduce the existing second-order extension for a one-dimensional flow, while providing a means for true multi-dimensional reconstruction. The new reconstruction is demonstrated in several verification studies. These include a shock-bubble interaction problem, an Euler-four-shock interaction problem, and the advection of a vortical disturbance. These results are presented, and the increased computation time and the effect of higher-order extension are discussed in this paper. The results show that the proposed multi-dimensional reconstruction provides a significant increase in the accuracy of the solution. We show that, despite the increase in the computational expense, the computational speed of the proposed QDS-N² method is several times higher than that of the previously proposed QDS-2N scheme for a fixed degree of numerical accuracy, at least, for the test problem of the advection of vertical disturbances.