An Efficient High-Order Gas-Kinetic Scheme (I): Euler equations

TL;DR

This paper introduces an efficient high-order gas-kinetic scheme (EHGKS) that improves computational efficiency and accuracy for solving Euler equations in compressible flows by simplifying the scheme and directly computing derivatives of macroscopic quantities.

Contribution

The paper develops a new high-order gas-kinetic scheme that extends existing methods with simplified computations and direct derivative calculations, enhancing efficiency and accuracy.

Findings

EHGKS achieves third, fifth, and seventh-order accuracy.

The scheme demonstrates good resolution of discontinuities and flow details.

EHGKS outperforms original HGKS and Runge-Kutta-WENO-GKS in efficiency.

Abstract

In this paper, an efficient high-order gas-kinetic scheme (EHGKS) is proposed to solve the Euler equations for compressible flows. We re-investigate the underlying mechanism of the high-order gas-kinetic scheme (HGKS) and find a new strategy to improve its efficiency. The main idea of the new scheme contains two parts. Firstly, inspired by the state-of-art simplifications on the third-order HGKS, we extend the HGKS to the case of arbitrary high-order accuracy and eliminate its unnecessary high-order dissipation terms. Secondly, instead of computing the derivatives of particle distribution function and their complex moments, we introduce a Lax-Wendroff procedure to compute the high-order derivatives of macroscopic quantities directly. The new scheme takes advantage of both HGKS and the Lax-Wendroff procedure, so that it can be easily extended to the case of arbitrary high-order accuracy with practical significance. Typical numerical tests are carried out by EHGKS, with the third, fifth and seventh-order accuracy. The presence of good resolution on the discontinuities and flow details, together with the optimal CFL numbers, validates the high accuracy and strong robustness of EHGKS. To compare the efficiency, we present the results computed by the EHGKS, the original HGKS and Runge-Kutta-WENO-GKS. This further demonstrates the advantages of EHGKS.