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

## TL;DR

This paper introduces a general synthetic iteration scheme (GSIS) that rapidly finds steady-state solutions for multiscale rarefied gas flows across all Knudsen numbers within dozens of iterations, surpassing existing methods.

## Contribution

The paper presents a novel, model-independent iterative scheme (GSIS) that significantly accelerates convergence for steady-state solutions in multiscale rarefied gas flows.

## Key findings

- GSIS achieves convergence within dozens of iterations across all Knudsen numbers.
- The method is extendable to mixture flows and chemically reacting flows.
- GSIS can enhance the efficiency of Monte Carlo simulations for near-continuum flows.

## 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 iteration 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 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 iteration scheme (GSIS) to find the steady-state solutions of general rarefied gas flows within dozens of iterations at any Knudsen number. As the GSIS does not rely on the specific kinetic model/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 also expected to accelerate the slow convergence in simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.