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.


Introduction
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 [1] 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 [4] 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  = 0 and  = 0 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 [7].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 zeropressure gas dynamics with the energy conservation law were defined in [10].
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 2 Advances in Mathematical Physics 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][12][13][14][15][16][17][18][19][20][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 2 × 2 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 quasilinear hyperbolic equations.And the results on interactions are also touchstones for the numerical schemes.Specifically, Sheng and Zhang [18] 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. [24] studied the interactions among delta-shock waves, vacuums, and contact discontinuities.In addition, with the help of a generalized plane wave solution, Yang [25] 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][27][28][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  01 ,  02 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.
For the case  − ≤  + , the solution containing two contact discontinuities and a vacuum state besides two constants is expressed as where () 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 (, , ; ,   , ℎ) be the delta shock solution of the form and then the following generalized Rankine-Hugoniot relation holds where [] =  − −  + .In order to ensure the uniqueness, the delta shock wave should satisfy the entropy condition which means that the characteristics on both sides of the discontinuity are in-coming.Under the entropy condition (7), by solving the ordinary differential equations ( 6) with the initial data  = 0: (0) = 0, (0) = 0, ℎ(0) = 0,   (0) = 0, one has For convenience, we now consider a special case when a delta shock wave is emitted at the beginning with the initial data satisfying  − >  0 >  + .It yields from ( 6) and (9) that One can check that the delta shock solution (10) satisfies the following: (1)   () is a monotone function of . ( While if  − =  + = 0, then   =  0 .(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: Case 1 ( − >   >  + ).In this case, two delta shock waves  1 and  2 will be emitted from ( 01 , 0) and ( 02 , 0), 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  + <   2 <   <   1 <  − by entropy condition (6), which means that  1 will overtake  2 at a finite time.The intersection point ( 0 ,  0 ) is calculated by which yields that At the intersection ( 0 ,  0 ), the new initial data are formed as follows: satisfying   1 >  0  >   2 .In view of  − >  + , a new delta shock wave will generate after interaction and we denote it with :  = ().The trajectory, velocity, and weights ((),   (), (), ℎ()) 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  1 determined by ( 12) is emitted from ( 01 , 0) and two contact discontinuities  1 :  =   and  2 :  =  + with a vacuum in between are emitted from ( 02 , 0), as shown in Figure 2.
Since the propagating speed of  1 satisfies   <   1 <  − , so  1 must meet the contact discontinuity  1 :  =   at  1 = ( 02 −  01 )/(  1 −   ), and a new delta shock wave  2 :  =  2 () forms, which is subjected to the generalized Rankine-Hugoniot relation with the initial data Therefore, by solving ( 18) and ( 19), we have It is clear that  2 will cross the vacuum region with a varying propagation speed.Noting that lim →+∞   2 () =  − >  + , so  2 will penetrate over the whole vacuum region and then meet  2 :  =  + at a finite time.The intersection ( 2 ,  2 ) is determined by At ( 2 ,  2 ), 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  3 after the interaction of  2 and  2 .
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  = 0 on the (, )-plane, as shown in Figure 3.

Vac Vac
Figure 4: The delta shock wave  1 collides with  1 at first and a new delta shock wave  2 generates.However, since lim →+∞   2 () =  − <  + ,  2 cannot penetrate over the vacuum region and finally has  2 () =  −  +  02 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 ( 01 , 0) and ( 02 , 0), respectively.Noting that the contact discontinuities  2 and  3 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.

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 [31] with 300 × 300 cells and CFL = 0.475.In what follows, by taking  01 = −0.2 and  02 = 0.2, we simulate the interaction of waves by four cases.For convenience, each situation will be simulated at two different times.
Case 1 ( − >   >  + ).We take the initial data as follows: The numerical results are presented by Figures 5-7.We observe from Figures 5-7 that when  = 1, two delta shock waves appear at (−0.2, 0) and (0.2, 0), respectively.As  increases, they will overtake each other and finally unify into a new delta shock wave at  = 6.5.
Case 2 ( − >  + >   ).We choose the following initial data The numerical results are shown in Figures 8-10.
From Figures 8-10, we can clearly see that, at  = 0.8, a delta shock wave and two contact discontinuities with a vacuum state in between are emitted from (−0.2, 0) and (0.2, 0), respectively.However, at  = 2.9, the delta shock wave penetrates over the whole vacuum region, and a new delta shock wave generates.
The numerical results are shown in Figures 11-13.Figures 11-13 imply that a delta shock wave is emitted from (−0.2, 0), and two contact discontinuities with a vacuum in between are emitted from (0.2, 0) at  = 0.5.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.

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Case 4 ( + >   >  − ).We select the initial data to be The numerical results are presented by Figures 14-16.
From Figures 14-16, we observe that both the contact discontinuities with a vacuum state in between are emitted from (−0.2, 0) and (0.2, 0) at  = 0.8, 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 t