The Perturbed Riemann Problem for a Geometrical Optics System

doi:10.1007/s42967-022-00192-3

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (3): 1148-1179.doi: 10.1007/s42967-022-00192-3

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The Perturbed Riemann Problem for a Geometrical Optics System

Shiwei Li, Hanchun Yang

Department of Mathematics, Yunnan University, Kunming, 650091, Yunnan, China

收稿日期:2021-10-31 修回日期:2022-03-16 出版日期:2023-09-20 发布日期:2023-08-29
通讯作者: Hanchun Yang,E-mail:hyang@ynu.edu.cn E-mail:hyang@ynu.edu.cn
基金资助:
The authors thank the referees for their valuable comments and suggestions to improve the presentation of the paper. This work is supported by the National Natural Science Foundation of China (12061084) and the Natural Science Foundation of Yunnan Province (2019FY003007).

The Perturbed Riemann Problem for a Geometrical Optics System

Shiwei Li, Hanchun Yang

Department of Mathematics, Yunnan University, Kunming, 650091, Yunnan, China

Received:2021-10-31 Revised:2022-03-16 Online:2023-09-20 Published:2023-08-29
Contact: Hanchun Yang,E-mail:hyang@ynu.edu.cn E-mail:hyang@ynu.edu.cn
Supported by:
The authors thank the referees for their valuable comments and suggestions to improve the presentation of the paper. This work is supported by the National Natural Science Foundation of China (12061084) and the Natural Science Foundation of Yunnan Province (2019FY003007).

摘要/Abstract

摘要： The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved. By studying the interactions among of the delta-shock, vacuum, and contact discontinuity, fourteen kinds of structures of Riemann solutions are obtained. The compound wave solutions consisting of delta-shocks, vacuums, and contact discontinuities are found. The single and double closed vacuum cavitations develop in solutions. Furthermore, it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data. Finally, the numerical results completely coinciding with theoretical analysis are presented.

关键词: Perturbed Riemann problem, Geometrical optics, Delta-shock wave, Vacuum, Compound wave, Numerical results

Abstract: The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved. By studying the interactions among of the delta-shock, vacuum, and contact discontinuity, fourteen kinds of structures of Riemann solutions are obtained. The compound wave solutions consisting of delta-shocks, vacuums, and contact discontinuities are found. The single and double closed vacuum cavitations develop in solutions. Furthermore, it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data. Finally, the numerical results completely coinciding with theoretical analysis are presented.

Key words: Perturbed Riemann problem, Geometrical optics, Delta-shock wave, Vacuum, Compound wave, Numerical results

中图分类号:

Shiwei Li, Hanchun Yang. The Perturbed Riemann Problem for a Geometrical Optics System[J]. Communications on Applied Mathematics and Computation, 2023, 5(3): 1148-1179.

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