# Unified Gas-kinetic Scheme with Multigrid Convergence for Rarefied Flow Study

## TL;DR

This paper introduces a multigrid accelerated implicit unified gas kinetic scheme (MIUGKS) that significantly improves computational efficiency for simulating rarefied and high-speed flows, outperforming traditional methods like DSMC.

## Contribution

The paper presents the first integration of geometric multigrid techniques into the implicit UGKS, greatly enhancing convergence speed for various flow regimes.

## Key findings

- MIUGKS achieves 5 to 9 times efficiency increase over previous implicit UGKS.
- MIUGKS is several orders of magnitude faster than DSMC for microflows.
- Even at hypersonic speeds, MIUGKS is over 100 times faster than DSMC.

## Abstract

The unified gas kinetic scheme (UGKS) is a direct modeling method based on the gas dynamical model on the mesh size and time step scales. With the implementation of particle transport and collision in a time-dependent flux function, the UGKS can recover multiple flow physics from the kinetic particle transport to the hydrodynamic wave propagation. In comparison with direct simulation Monte Carlo (DSMC), the equations-based UGKS can use the implicit techniques in the updates of macroscopic conservative variables and microscopic distribution function. The implicit UGKS significantly increases the convergence speed for steady flow computations, especially in the highly rarefied and near continuum regime. In order to further improve the computational efficiency, for the first time a geometric multigrid technique is introduced into the implicit UGKS, where the prediction step for the equilibrium state and the evolution step for the distribution function are both treated with multigrid acceleration. The multigrid implicit UGKS (MIUGKS) is used in the non-equilibrium flow study, which includes microflow, such as lid-driven cavity flow and the flow passing through a finite-length flat plate, and high speed one, such as supersonic flow over a square cylinder. The MIUGKS shows 5 to 9 times efficiency increase over the previous implicit scheme. For the low speed microflow, the efficiency of MIUGKS is several orders of magnitude higher than the DSMC. Even for the hypersonic flow at Mach number 5 and Knudsen number 0.1, the MIUGKS is still more than 100 times faster than the DSMC method for a convergent steady state solution.