Riemann Problem and Wave Interactions for a Class of Strictly Hyperbolic Systems of Conservation Laws
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Riemann Problem and Wave Interactions for a Class of Strictly Hyperbolic Systems of Conservation Laws Yu Zhang1 · Yanyan Zhang2 Received: 5 May 2019 / Accepted: 19 November 2019 © Sociedade Brasileira de Matemática 2019
Abstract A class of strictly hyperbolic systems of conservation laws are proposed and studied. Firstly, the Riemann problem with initial data of two piecewise constant states is constructively solved. The solutions involving contact discontinuities and delta shock waves are obtained. The generalized Rankine–Hugoniot relation and entropy condition for the delta shock wave are clarified and the existence and uniqueness of the deltashock solution is proved. Furthermore, the global structure of solutions with five different configurations is constructed via investigating the interactions of delta shock waves and contact discontinuities. Finally, we present a typical example to illustrate the application of the system introduced. Keywords Strictly hyperbolic systems of conservation laws · Riemann problem · Delta shock wave · Contact discontinuity · Wave interactions Mathematics Subject Classification 35L65 · 35L67 · 35L60
Supported by National Science Foundation of China (11501488), Yunnan Applied Basic Research Projects (2018FD015), the Scientific Research Foundation Project of Yunnan Education Department (2018JS150) and Nan Hu Young Scholar Supporting Program of XYNU.
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Yu Zhang [email protected] Yanyan Zhang [email protected]
1
Department of Mathematics, Yunnan Normal University, Kunming 650500, People’s Republic of China
2
College of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, People’s Republic of China
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Y. Zhang, Y. Zhang
1 Introduction The Aw–Rascle model for traffic flow in the conservative form can be expressed as
ρ (ρu)x = 0, t + ρ u + p(ρ) + ρu u + p(ρ) = 0, t
(1)
x
where ρ ≥ 0 and u ≥ 0 denote the density and velocity of cars on the roadway, respectively. The “pressure” p takes the form p(ρ) = ρ γ with γ > 0, which plays the role of an anticipation factor, taking into account the drivers’ reaction to the state of traffic in front of them. The system of conservation laws (1) has received extensive attention and has been widely applied to study the formation and dynamics of traffic jams. It was initially proposed by Aw and Rascle (2000) to remedy the deficiencies of second order models of car traffic pointed out by Daganzo (1995) and had also been independently derived by Zhang (2002). In 2013, Pan and Han (2013) considered the Chaplygin gas pressure (Chaplygin 1904), whose equation of state is p(ρ) = −/ρ with constant > 0. Then, (1) is transformed into ρt + (ρu)x = 0, (2) (ρu)t + (ρu 2 − u)x = 0. The Riemann problem was solved. Besides, the authors proved that, as pressure vanishes, namely, as → 0, the Riemann solutions of (2) converge to those of the following zero-pressure flow
ρt + (ρu)x = 0, (ρu)t + (ρu 2 )x = 0,
(3)
which is also named the transport equations or pressureless gas dynamics model, and usually used to d
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