Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law
We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown.
As is well known, the system of zero-pressure gas dynamics consisting of conservation laws of mass and momentum, which is also called the transport equations, or Euler equations for pressureless fluids, has been extensively investigated since the 90s of 20th century. It is derived from Boltzmann equations  and the flux-splitting scheme of the full compressible Euler equations [2, 3] and can be used to describe the motion process of free particles sticking together under collision  and the formation of large-scale structures in the universe [5, 6].
However, we have to mention that, as having no pressure, the energy transport must be taken into account for the considered media. Therefore, it is very necessary to consider the conservation law of energy in zero-pressure gas dynamics. To this end, we study the one-dimensional zero-pressure gas dynamics governed by the conservation laws of mass, momentum, and energy:where and represent the density and velocity, respectively, is the internal energy and assumed to be nonnegative, and is the internal energy per unit mass. The regions in the physical space where and are identified with the vacuum regions of the flow. Here, is considered as an independent variable just for convenience.
System (1) was early studied by Kraiko . In contrast to the traditional zero-pressure gas dynamics system that contains only the conservation laws of mass and momentum, in order to construct the solution of (1) for arbitrary initial data, a new type of discontinuities which are different from the classical ones and carry mass, impulse, and energy are needed. In [8, 9], system (1) was further discussed. Some special integral identities were introduced to define the delta-shock solutions and construct the Rankine-Hugoniot relation for delta shock waves. Moreover, using these integral identities, the balance laws describing mass, momentum, and energy transport from the area outside the delta shock wave front onto its front were derived. What is more, the delta shock wave type solutions for multidimensional zero-pressure gas dynamics with the energy conservation law were defined in .
A delta shock wave is a generalization of an ordinary shock wave. Roughly speaking, it is a kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its support. It is more compressive than an ordinary shock wave and more characteristics enter the discontinuity line. Physically, the delta shock waves describe the process of formation of the galaxies in the universe and the process of concentration of particles. As for delta shock waves, there are numerous excellent papers, see [11–21] and so forth. Nevertheless, compared to these results, a distinctive feature for (1) is that the Dirac delta functions develop in both state variables and simultaneously, which is quite different from those aforementioned, in which only one state variable contains the Dirac delta function. In fact, the theory of delta shock waves with Dirac delta functions developing in both state variables has been established by Yang and Zhang [22, 23] for a class of nonstrictly hyperbolic systems of conservation laws.
In the past over two decades, the investigation of interactions of delta shock waves has been increasingly active. This is important not only because of their significance in practical applications but also because of their basic role as building blocks for the general mathematical theory of quasi-linear hyperbolic equations. And the results on interactions are also touchstones for the numerical schemes. Specifically, Sheng and Zhang  discussed the overtaking of delta shock waves and vacuum states in one-dimensional zero-pressure gas dynamics. By solving the two-dimensional Riemann problems for zero-pressure gas dynamics with three constant states, Cheng et al.  studied the interactions among delta-shock waves, vacuums, and contact discontinuities. In addition, with the help of a generalized plane wave solution, Yang  studied a type of generalized plane delta-shock wave for the -dimensional zero-pressure gas dynamics and investigated the overtaking of two plane delta shocks. For more works on the interactions of delta shock waves, we refer to [26–29] and so forth.
Motivated by the discussions above, in the present paper, we are concerned with the interactions among delta shock waves, vacuum states, and contact discontinuities in solutions. Therefore, we study the Riemann problem of (1) with initial data of three piecewise constant states as follows:where , , are arbitrary constants and , are any two fixed points on -axis.
We will deal with the Riemann problem (1), (2) case by case along with constructing the solutions. For this purpose, it is necessary to consider whether two adjacent waves intersect and interact with each other when constructing the global solution. However, it is often not so easy to see whether two delta shock waves meet and how they interact with each other. Therefore, some technical treatments are needed.
This paper is arranged as follows. In Section 2, the delta shock solution of (1) is reviewed and a general case when the delta shock wave is emitted at the beginning with a nonzero initial data is considered. Section 3 discusses the interactions of the delta shock waves and vacuum states. The Riemann solutions of (1), (2) are constructed globally. Finally, four kinds of numerical simulations coinciding with the theoretical analysis are presented in Section 4.
2. Delta-Shock Solution
System (1) has a triple eigenvalue and two right eigenvectors and . Since for , is linearly degenerate, which means that the elementary waves involve only contact discontinuities. The self-similar solution is constructed by two cases.
For the case , the solution containing two contact discontinuities and a vacuum state besides two constants is expressed aswhere is a smooth function satisfying and .
For the case , the singularity of solutions must develop because of the overlap of characteristic lines. Therefore, the solution involving a delta shock wave is introduced.
Let be the delta shock solution of the formand then the following generalized Rankine-Hugoniot relation holdswhere . In order to ensure the uniqueness, the delta shock wave should satisfy the entropy conditionwhich means that the characteristics on both sides of the discontinuity are in-coming.
For convenience, we now consider a special case when a delta shock wave is emitted at the beginning with the initial datasatisfying It yields from (6) and (9) thatOne can check that the delta shock solution (10) satisfies the following: (1) is a monotone function of . (2)If , then While if , then (3)
3. Interactions of Delta Shock Waves
In this section, we analyze the interactions of delta shock waves. To ensure that all the cases are covered completely, according to the relation among , our discussion is divided into four cases:(1);(2);(3);(4).
Case 1 (). In this case, two delta shock waves and will be emitted from and , respectively, as shown in Figure 1.
According to what has been discussed in Section 2, these two delta shock waves are uniquely determined by
We have by entropy condition (6), which means that will overtake at a finite time. The intersection point is calculated bywhich yields that
At the intersection , the new initial data are formed as follows:satisfying .
In view of , a new delta shock wave will generate after interaction and we denote it with . The trajectory, velocity, and weights of can be uniquely obtained by solving the ordinary differential equations (6) with the initial date (16). The detail is omitted.
Thus, the result of interaction of two delta shock waves is still a single delta shock wave. This fact can be formulated as
Case 2 ( (when , the structure of solution is similar)). In this situation, a delta shock wave determined by (12) is emitted from and two contact discontinuities and with a vacuum in between are emitted from , as shown in Figure 2.
Since the propagating speed of satisfies , so must meet the contact discontinuity at , and a new delta shock wave forms, which is subjected to the generalized Rankine-Hugoniot relationwith the initial data
It is clear that will cross the vacuum region with a varying propagation speed. Noting that , so will penetrate over the whole vacuum region and then meet at a finite time. The intersection is determined by
At , a new initial value problem is formed and can be solved similar to Case 1. We denote the delta shock wave connecting two constant states and with after the interaction of and .
The conclusion of this case is that the delta shock wave will penetrate over the whole vacuum region between two contact discontinuities. This fact is expressed as
Case 3 ((when , the structure of solution is similar)). Similar to Case 2, there are a delta shock wave, two contact discontinuities, and a vacuum near on the -plane, as shown in Figure 3.
The delta shock wave collides with at first and a new delta shock wave generates. However, since , cannot penetrate over the vacuum region and finally has as its asymptote. This fact is symbolized as
Case 4 (). In this situation, both the contact discontinuities with a vacuum state in between are emitted from and , respectively. Noting that the contact discontinuities and own the same propagating speed, thus there is no collision of waves and the solution is expressed as which is called a collisionless solution, as shown in Figure 4.
4. Numerical Simulations
In order to verify the validity of the interactions of delta shock waves and vacuum states mentioned in Section 3, we present some representative numerical simulations in this section. Many more numerical tests have been performed to make sure that what are presented are not numerical artifacts.
To discretize the system, we employ the second-order nonoscillatory central schemes  with cells and . In what follows, by taking and , we simulate the interaction of waves by four cases. For convenience, each situation will be simulated at two different times.
From Figures 8–10, we can clearly see that, at , a delta shock wave and two contact discontinuities with a vacuum state in between are emitted from and , respectively. However, at , the delta shock wave penetrates over the whole vacuum region, and a new delta shock wave generates.
Figures 11–13 imply that a delta shock wave is emitted from , and two contact discontinuities with a vacuum in between are emitted from at . But the delta shock wave can not penetrate over the whole vacuum region even though time is on the increase. In this process, the region of vacuum state keeps expanding.
From Figures 14–16, we observe that both the contact discontinuities with a vacuum state in between are emitted from and at , respectively. As time goes on, the vacuum state keeps continuously expanding and never disappears.
To sum up, all of the above numerical results clearly reveal the interactions of delta shock waves and vacuum states discussed in Section 3. We also indicate that because of the occurrence of singularity as the weighted Dirac delta functions, some oscillations appear in the numerical experiments as shown in Figures 8 and 11. It may be a challenge for numerical schemes when delta shock waves develop in solutions.
The authors declare that they have no competing interests.
This work is supported by the National Natural Science Foundation of China (No. 11501488), the Scientific Research Foundation of Xinyang Normal University (No. 0201318), and Nan Hu Young Scholar Supporting Program of XYNU.
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