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

In this paper, a true-direction flux reconstruction of the 2nd order quiet direct simulation (QDS) as an equivalent Euler equation solver, named QDS-N², is proposed. Due to 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 thus 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 flux computation, QDS particles are introduced using properties based on weighted moments taken over the polynomial reconstruction of conserved quantity fields. The resulting flux expressions are shown to exactly reproduce existing second-order extension for one-dimensional flow, while providing a means for true multi-dimension reconstruction. The new reconstruction is demonstrated in two verification cases. These include a shock bubble 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. Results show that our proposed multi-dimensional reconstruction provides a significant increase in the accuracy of the solution. We show that, despite the increase in computational expense, the proposed QDS-N² method is several times faster than the previously proposed QDS-2N scheme for a fixed degree of numerical accuracy.