Limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type.

Lihui Guo,Gan Yin

doi:10.1155/2014/287256

Abstract

The limit of Riemann solutions to the nonsymmetric system of Keyfitz-Kranzer type with a scaled pressure is considered for both polytropic gas and generalized Chaplygin gas. In the former case, the delta shock wave can be obtained as the limit of shock wave and contact discontinuity when u − > u + and the parameter ϵ tends to zero. The point is, the delta shock wave is not the one of transport equations, which is obviously different from cases of some other systems such as Euler equations or relativistic Euler equations. For the generalized Chaplygin gas, unlike the polytropic or isothermal gas, there exists a certain critical value ϵ 2 depending only on the Riemann initial data, such that when ϵ drops to ϵ 2, the delta shock wave appears as u − > u +, which is actually a delta solution of the same system in one critical case. Then as ϵ becomes smaller and goes to zero at last, the delta shock wave solution is the exact one of transport equations. Furthermore, the vacuum states and contact discontinuities can be obtained as the limit of Riemann solutions when u − < u + and u − = u +, respectively.

Highlights

The nonsymmetric system of Keyfitz-Kranzer type can be written as ρt + (ρφ (ρ, u1, u2, . . . , un))x = 0, (1)(ρui)t + (ρuiφ (ρ, u1, u2, . . . , un))x, i = 1, 2, . . . , n, where φ (ρ, u) = φ (u) − p (ρ) (2)is a nonlinear function
In 2008, Berthelin et al [3] studied the limit behavior which was investigated by changing p into εp and taking p(ρ) = (1/ρ − 1/ρ∗), ρ ≤ ρ∗, where ρ∗ is the maximal density which corresponds to a total traffic jam and is assumed to be a fixed constant it should depend on the velocity in practice
We focus on system (9) with equation of state for both polytropic gas and generalized Chaplygin gas

Summary

Introduction

The propagating speed and the strength of the delta shock wave in the limit situation are different from the classical results of transport equations (8) with the same Riemann initial data. Given any two constant states (u±, ρ±), we can constructively obtain the Riemann solutions of (8) and (12) containing contact discontinuities, vacuum, or delta shock wave. Suppose that Ω ⊂ R × R+ is a region cut by a smooth curve Γ = {(x, t) | x = x(t)} into left and right hand parts Ω± = {(x, t) | ±(x − x(t)) > 0}; (u(x, t), ρ(x, t)) is a generalized δ-shock wave solution of system (9) and (11); functions ρ(x, t) and u(x, t) are smooth in Ω± and have one-side limits ρ±, u± on the curve Γ. We have obtained the solutions of the Riemann problem for (9)

Limit of Riemann Solutions to the KeyfitzKranzer Type System

Conclusion