Flux-Approximation Limits of Solutions to the Brio System with Two Independent Parameters
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Flux-Approximation Limits of Solutions to the Brio System with Two Independent Parameters Yanyan Zhang1
· Yu Zhang2
Received: 3 March 2019 / Accepted: 11 March 2020 © Springer Nature B.V. 2020
Abstract By the flux-approximation method, we study limits of Riemann solutions to the Brio system with two independent parameters. The Riemann problem of the perturbed system is solved analytically, and four kinds of solutions are obtained constructively. It is shown that, as the two-parameter flux perturbation vanishes, any two-shock-wave and tworarefaction-wave solutions of the perturbed Brio system converge to the delta-shock and vacuum solutions of the transport equations, respectively. In addition, we specially pay attention to the Riemann problem of a perturbed simplified system of conservation laws derived from the perturbed Brio system by neglecting some quadratic term. As one of the parameters of the perturbed Brio system goes to zero, the solution of which consisting of two shock waves tends to a delta-shock solution to this simplified system. By contrast, the solution containing two rarefaction waves converges to a contact discontinuity and a rarefaction wave of the simplified system. What is more, the formation mechanisms of delta shock waves under flux approximation with both two parameters and only one parameter are clarified. Some numerical simulations presenting the formation processes of delta shock waves and vacuum states are also presented to confirm the theory analysis.
Keywords Brio system · Transport equations · Riemann problem · Delta shock wave · Vacuum · Flux approximation · Numerical simulations
Supported by National Natural Science Foundation of China (11501488, 11801490), the Scientific Research Foundation of Xinyang Normal University (No. 0201318), Nan Hu Young Scholar Supporting Program of XYNU, Yunnan Applied Basic Research Projects (2018FD015) and the Scientific Research Foundation Project of Yunnan Education Department (2018JS150).
B Y. Zhang
[email protected]
1
College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, P.R. China
2
Department of Mathematics, Yunnan Normal University, Kunming 650500, P.R. China
Y. Zhang, Y. Zhang
1 Introduction As to our knowledge, in the past over two decades, the delta shock wave has been systematically studied by a large number of scholars. For example, see the results in [10, 12, 20–24, 27, 28] and the references cited therein. Particularly, in the related researches of delta shock waves, one of the interesting topics is to explore the formation of delta shock waves and vacuum states in solutions, which correspond to the phenomena of concentration and cavitation, respectively. At this moment, an effective approach is to use the so called vanishing pressure limit method, which was early proposed by Chen and Liu [6, 7] to study the formation of delta shock waves and vacuums for the Euler equations of isentropic and nonisentropic gas dynamics, respectively. See also Li [13] for the isothermal Euler equations with zero tempe
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