# Comparative Study on Flux Solution Methods of Discrete Unified Gas Kinetic Scheme

## TL;DR

This paper compares different flux calculation methods in a gas kinetic scheme, focusing on their accuracy and efficiency in various flow simulations.

## Contribution

The paper introduces the Simpson method for flux calculation in DUGKS and compares its performance with other variants.

## Key findings

- Simpson–DUGKS reduces computational time by about 59% compared to original DUGKS under certain conditions.
- Optimized DUGKS shows the smallest L2 error for velocity in Taylor–Green vortex flow but the largest for density.
- Optimized DUGKS results deviate more from reference solutions in lid-driven cavity flow with small grid numbers.

## Abstract

In this work, the Simpson method is proposed to calculate the interface flux of a discrete unified gas kinetic scheme (DUGKS) according to the distribution function at the node and the midpoint of the interface, which is noted by Simpson–DUGKS. Moreover, the optimized DUGKS and Simpson–DUGKS considering the force term are derived. Then, the original DUGKS, optimized DUGKS, and Simpson–DUGKS are compared and analyzed in theory. Finally, the numerical tests are performed under different grid numbers (N). In the steady unidirectional flow (Couette flow and Poiseuille flow), the three methods are stable under different Courant–Friedrichs–Lewy (CFL) numbers, and the calculated L2 errors are the same. In the Taylor–Green vortex flow, the L2 error of the optimized DUGKS is the smallest with respect to the analytical solution of velocity, but the L2 error of the optimized DUGKS is the largest with respect to the analytical solution of density. In the lid-driven cavity flow, the results of the optimized DUGKS deviate more from the reference results in terms of accuracy, especially in the case of a small grid number. In terms of computational efficiency, it should be noted that the computational time of optimized DUGKS increases by about 40% compared with the original DUGKS when CFL = 0.1 and N = 16, and the calculation time of Simpson–DUGKS is reduced by about 59% compared with the original DUGKS when CFL = 0.95 and N = 16.