The Riemann solutions comparison for compressible Euler equations in different Chaplygin gas states

Abstract

In this article, we make several comparisons of Riemann solutions of compressible Euler equations in the modified Chaplygin gas state ($p=A\rho -{\rho }^{-1}$) and classical Chaplygin gas state ($p=-{\rho }^{-1}$). Firstly, through the analysis of characteristic method, we construct the Riemann solutions for the modified Chaplygin gas state and compare the Riemann solutions with different coefficients A. Then we obtain the reason of the occurring of delta shock waves in classical Chaplygin gas case, which is the lower bounded of 1-wave curves and the upper bounded of 3-wave curves. Moreover, by comparing the position of simple wave curves between the modified and classical Chaplygin gas state, we find that no matter how small the coefficient A is, there is an initial Riemann data, subject to that the Riemann solutions for modified Chaplygin gas state contain rarefaction waves while the Riemann solutions for classical Chaplygin gas state are delta waves, which is different from the initial expect. Finally, we present a numerical analysis consistent with the theoretical analysis.

COMMUNICATED BY:

• D. Cao

Keywords:

• Comparison of Riemann solutions
• Chaplygin gas
• non-isentropic

2010 Mathematics Subject Classification:

• 35L65

Acknowledgments

The authors thank the reviewers for he/her useful comments, which can improve our results and expression. Weifeng Jiang would also like to thank the kindness of Prof. Tong Li, Prof. Lihe Wang and the support of Department of Mathematics in University of Iowa during 2020–2021.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the Natural Science Foundation of Zhejiang (Grant No. LQ18A010004), China Scholarship Council Funding (Grant No. 201908330112), Scientific research program of Hubei Provincial Department of Education(Grant No.Q20201602) and Wenzhou Science and Technology Bureau (Grant No.2020G0038).