A simplification of the unified gas kinetic scheme

TL;DR

This paper introduces a simplified version of the unified gas kinetic scheme (UGKS) that maintains high accuracy while reducing computational costs to levels comparable with the discrete ordinate method, enhancing efficiency in transition flow simulations.

Contribution

A new simplified UGKS that evaluates the equilibrium flux analytically, reducing computational cost without sacrificing accuracy, and demonstrating its Navier-Stokes asymptotic preserving property.

Findings

Simplified scheme matches UGKS accuracy with lower computational cost.

Equilibrium flux can be evaluated analytically, avoiding velocity space discretization.

The scheme remains Navier-Stokes asymptotic preserving when collision effects are included.

Abstract

Unified gas kinetic scheme (UGKS) is an asymptotic preserving scheme for the kinetic equations. It is superior for transition flow simulations, and has been validated in the past years. However, compared to the well known discrete ordinate method (DOM) which is a classical numerical method solving the kinetic equations, the UGKS needs more computational resources. In this study, we propose a simplification of the unified gas kinetic scheme. It allows almost identical numerical cost as the DOM, but predicts numerical results as accurate as the UGKS. Based on the observation that the equilibrium part of the UGKS fluxes can be evaluated analytically, the equilibrium part in the UGKS flux is not necessary to be discretized in velocity space. In the simplified scheme, the numerical flux for the velocity distribution function and the numerical flux for the macroscopic conservative quantities are evaluated separately. The simplification is equivalent to a flux hybridization of the gas kinetic scheme for the Navier-Stokes (NS) equations and conventional discrete ordinate method. Several simplification strategies are tested, through which we can identify the key ingredient of the Navier-Stokes asymptotic preserving property. Numerical tests show that, as long as the collision effect is built into the macroscopic numerical flux, the numerical scheme is Navier-Stokes asymptotic preserving, regardless the accuracy of the microscopic numerical flux for the velocity distribution function.