- Research
- Open Access
The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws
- Guodong Wang^{1},
- Jia-Bao Liu^{1}Email author and
- Lin Zhao^{1}
- Received: 21 March 2017
- Accepted: 27 June 2017
- Published: 14 July 2017
Abstract
The Riemann problem for a one-dimensional nonlinear wave system with different gamma laws is considered. By the properties of wave curves, we observe that this system does not contain the composite wave compared to the barotropic models of gas dynamics with different pressure laws. Under some initial value data, the Riemann solution is constructed. Using the interaction of the elementary waves, we consider the generalized Riemann problem and discover that the Riemann solution is stable for such perturbation of the initial data.
Keywords
- one-dimensional nonlinear wave system
- different gamma laws
- Riemann problem
- perturbed initial data
MSC
- 35L65
- 35J70
- 35R35
1 Introduction
The organization of this paper is as follows. In Section 2, we describe the properties of wave curves and construct the Riemann solution. In Section 3, we consider the initial value problem with three constant states. By the interaction between the stationary contact wave and the shock wave or rarefaction wave, the global solutions are constructed. Moreover, we obtain that the solution of the perturbed initial value problem converges to the corresponding Riemann solution as ε approaches zero, which shows the stability of the Riemann solution for the small perturbation.
2 The solution to the Riemann problem
In order to analyze the solutions to system (1.3), we need to look at the wave curves.
2.1 Wave curves
Remark 1
As a consequence, the \(\gamma(x)\) remains constant across a rarefaction wave or a shock wave and only changes along the contact wave. In addition, the shock speed σ does not vanish, i.e., there is not a stationary shock. So system (1.3) does not contain the curves of composite waves [22–25].
2.2 The properties of the elementary waves
To solve (1.3) and (1.4), we project all the wave curves on the \((\rho,m)\)-plane. Now, let us investigate the properties of the wave curves.
Lemma 2.1
The curve \(R_{1}(U_{l},U)\) is monotonic decreasing and concave, while \(R_{3}(U_{r},U)\) is monotonic increasing and convex.
Proof
Lemma 2.2
The curve \(S_{1}(U_{l},U)\) is monotonic decreasing and concave, while \(S_{3}(U,U_{r})\) is monotonic increasing and convex.
Proof
Here, we only prove the result for the case \(S_{1}(U_{l},U)\), and the other case can be studied in a similar method.
Similar arguments lead to the result that \(S_{3}(U_{r},U)\) is monotonic increasing and convex.
Therefore, we complete the proof. □
Let the one-rarefaction wave \(R_{1}(U,U_{0})\) and the one-shock wave \(S_{1}(U,U_{0})\) (or the three-rarefaction wave \(R_{3}(U,U_{0})\) and the three-shock wave \(S_{3}(U,U_{0})\)) pass through the point \(P_{0}=(\rho_{0},m_{0},\gamma_{l})\) respectively, we have the following lemma.
Lemma 2.3
The curves \(R_{1}(U,U_{0})\) (\(R_{3}(U,U_{0})\)) contact with \(S_{1}(U,U_{0})\) (\(S_{3}(U,U_{0})\)) at point \(P_{0}\) up to the second order, respectively.
Proof
Similar calculations show that the other result of the lemma is true. So, the proof is completed. □
2.3 The Riemann solution
In this paper, we only consider the case \(\gamma_{l}>\gamma_{r}\) and can obtain the corresponding results for the other case \(\gamma_{l}<\gamma_{r}\) in a similar way. For simplicity, we take \(A=1\) in (1.2).
Given the left state \(U_{l}=(\rho_{l},m_{l},\gamma_{l})\) and the right state \(U_{r}=(\rho_{r},m_{r},\gamma_{r})\), we consider the projection of the wave curves in the \((\rho,m)\)-plane. Let \(U_{-}=(\rho _{-},m_{-},\gamma_{l})\) and \(U_{+}=(\rho_{+},m_{+},\gamma_{r})\) satisfy that \(U_{-}\in R_{1}(U_{l},U)\) or \(S_{1}(U_{l},U)\), \(U_{+}\in R_{3}(U_{r},U)\) or \(S_{r}(U_{r},U)\) and Eq. (2.15). Denote \(w_{l}=m_{l}+\frac{2\sqrt{\gamma_{l}}}{1+\gamma_{l}}\rho_{l}^{\frac {1+\gamma_{l}}{2}}\) and \(z_{r}=m_{r}-\frac{2\sqrt{\gamma_{r}}}{1+\gamma_{r}}\rho_{r}^{\frac {1+\gamma_{r}}{2}}\).
We assume \(\rho_{-}>1\) and describe the Riemann solution \(\mathit{RP}(U_{l},U_{r})\) as the following five cases (for the other case \(\rho_{-}<1\), one easily obtains similar results).
In the following cases, we can obtain the uniqueness of the corresponding Riemann solution and omit the details.
Remark 2
We have demonstrated the corresponding Riemann solutions for the case \(\rho_{-}>1\). For the case \(0<\rho_{-}<1\), we note \(\rho_{-}>\rho_{+}\) for \(\gamma_{l}>\gamma_{r}\) and easily obtain the Riemann solutions by similar methods.
We have constructed the Riemann solutions for all the cases and have the following theorem.
3 Interaction between the stationary contact wave and the elementary waves
In this section, we only consider the interaction of the stationary contact wave with the rarefaction wave or the shock wave. As for the interaction between the rarefaction wave and the shock wave, we may see [14, 21] or other related results. Since the speed of one-wave (\(R_{1}\) or \(S_{1}\)) is less than zero and that of three-wave (\(R_{3}\) or \(S_{3}\)) is greater than zero, the interaction of the stationary contact wave with the shock or the rarefaction wave may be divided into four cases:
\(J_{2}+S_{1}\), \(J_{2}+R_{1}\), \(S_{3}+J_{2}\), \(R_{3}+J_{2}\).
Case 1. The collision of \(J_{2}\) and \(S_{1}\).
First, we consider the subcase \(\rho_{-}>1\), see Figure 6. It is clear that when \(J_{2}\) collides with \(S_{1}\) at some point, the new Riemann problem is formed. We claim that the corresponding Riemann solution \(\mathit{RP}(U_{l},U_{r})\) is \(U_{l}+S_{1}(U_{l},U_{-})+U_{-}+J_{2}(U_{-},U_{+})+U_{+}+S_{3}(U_{+},U_{r})+U_{r}\) and is not \(U_{l}+S_{1}(U_{l},U_{l'})+J_{2}(U_{l'},U_{r'})+R_{3}(U_{r'},U_{r})+U_{r}\).
In order to obtain the above statement, let us claim that \(\rho_{+}>\rho_{0'}\), where \(U_{0'}\in S_{1}(U_{0},U)\) and \(m_{-}=m_{0'}=m_{+}\). The reason is as follows.
If \(\rho_{l}<1\), we have \(\rho_{0}<1\) and also obtain similar results.
For the subcase \(\rho_{-}<1\), we can obtain similar results and omit the details here. In the following cases, we only consider the subcase \(\rho_{-}>1\).
Furthermore, we observe that as \(\varepsilon\rightarrow0\), the limit of the solution of (1.3) and (3.1) is the corresponding Riemann solution of (1.3) and (1.4).
Case 2. The collision of \(J_{2}\) and \(R_{1}\).
Moreover, it is clear that as \(\varepsilon\rightarrow0\), the solution of the initial value problem transforms into \(\mathit{RP}(U_{l},U_{r})\).
Case 3. The collision of \(S_{3}\) and \(J_{2}\).
If \(\rho_{-}>1\), which indicates \(\rho_{+}>1\), we have \(\rho_{r}>1\), which implies \(\rho_{0}>1\). If \(\rho_{-}<1\), we have \(\rho_{+}<1\), which implies \(\rho_{r}<1\).
As \(\varepsilon\rightarrow0\), the solution of the initial value problem reduces to \(\mathit{RP}(U_{l},U_{r})\).
Case 4. The collision of \(R_{3}\) and \(J_{2}\).
Now, we draw a one-shock wave \(S_{1}(U_{-},U_{l})\) from the state \(U_{l}\) and a three-rarefaction wave \(R_{3}(U_{+},U_{r})\) from the state \(U_{r}\). Here \(U_{\pm}\) satisfies that \(m_{-}=m_{+}\), \(m_{\pm}< m_{l}\) and \(m_{\pm}< m_{r}\). Then the Riemann solution \(\mathit{RP}(U_{l},U_{r})\) is \(U_{l}+S_{1}(U_{-},U_{l})+U_{-}+J_{2}(U_{-},U_{+})+U_{+}+R_{3}(U_{+},U_{r})+U_{r}\), see Figure 9.
In addition, it is not difficult to find that \(\mathit{RP}(U_{l},U_{r})\) is the limit of the solution of the initial value problem.
So far, we have discussed the interactions of the contact wave with the rarefaction wave or the shock wave and have constructed the solutions for the initial value problem (1.3) and (3.1) or (3.7). Therefore, we obtain the following theorem.
Theorem 3.1
There exists a unique solution to the perturbed initial value problem (1.3) and (3.1) or (3.7). The limit of the perturbed Riemann solution of (1.3) and (3.1) or (3.7) is exactly the corresponding Riemann solution of (1.3) and (1.4). The Riemann solution of (1.3) and (1.4) is stable with respect to such small perturbations of the initial data.
4 Concluding remarks
In this paper, we present the Riemann problem and the interactions of the stationary contact discontinuity with the elementary waves. We discover the stability of the generalized Riemann problem, but do not observe the composite wave, which motivates us to consider the related problems including the coupling of two different hyperbolic systems.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees who provided valuable comments and suggestions resulting in improvements in this manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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