General Implicit Iterative Method for Unified Gas-kinetic Scheme

TL;DR

This paper introduces a two-step implicit unified gas-kinetic scheme (IUGKS) that enhances computational efficiency for multi-scale flow simulations by accelerating convergence through implicit evaluation and incorporating the full Boltzmann collision term.

Contribution

The paper proposes a novel two-step IUGKS that improves efficiency and accuracy by implicit macroscopic variable evaluation and full Boltzmann collision integration.

Findings

Achieves faster convergence in multi-scale flow simulations.

Validates high accuracy across flow regimes from low speed to hypersonic.

Demonstrates superior efficiency compared to existing implicit schemes.

Abstract

In order to further enhance the computational efficiency of the implicit unified gas-kinetic scheme (IUGKS, JCP 315 (2016) 16-38) for multi-scale flow simulation, a two-step IUGKS is proposed in this paper. The multiscale solution of the UGKS is determined by the integral solution of the kinetic model equation, which is composed of the Lagrangian integration of the equilibrium and the free particle transport of the nonequilibrium state. With the implicit evaluation of the macroscopic variables in the first iterative step, the integration of the equilibrium can be directly used in the flux calculation of macroscopic flow variables in the second iterative step. This is equivalent to include the viscous flux in the implicit scheme to accelerate the convergence of the solution instead of using the Euler flux in the iterative process of the original IUGKS. In the present IUGKS, the update of macroscopic flow variables are closely coupled with the implicit evolution of the gas distribution function. At the same time, in order to get the more accurate solution, the full Boltzmann collision term is incorporated into the current scheme through the penalty method. Different iterative techniques, such as LU-SGS and multi-grid, are used for solving the linear algebraic system of coupled macroscopic and microscopic equations. The efficiency of the IUGKS has reached a favorable level among all implicit schemes for the kinetic equations in the literature. Several numerical examples are used to validate the performance of the IUGKS. Accurate solutions have been obtained efficiently in all flow regimes from low speed to hypersonic ones.