Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?

Abstract

One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iterative scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the “converged” solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the number of iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iterative scheme (GSIS) to find the steady-state solutions of rarefied gas flows within dozens of iterations at any Knudsen number. The key ingredient of our scheme is that the macroscopic equations, which are solved together with the Boltzmann equation and help to adjust the velocity distribution function, not only asymptotically preserve the Navier-Stokes limit in the framework of Chapman-Enskog expansion, but also contain the Newton's law for stress and the Fourier's law for heat conduction explicitly. For this reason, like the implicit UGKS, the constraint that the spatial cell size should be smaller than the mean free path of gas molecules is removed, but we do not need the complex evaluation of numerical flux at cell interfaces. What's more, as the GSIS does not rely on the specific collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are expected to accelerate the slow convergence in the simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.