Four-Quadrant Riemann Problem for a 2 × 2 System Involving Delta Shock

Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Division of Applied Mathematical Sciences, Korea University, Sejong 30019, Korea; jinahwang@korea.ac.kr (J.H.); angelic52@korea.ac.kr (S.S.) 2 Department of Ocean Systems Engineering, Sejong University, Seoul 05006, Korea; Myoungin@sju.ac.kr * Correspondence: woonjae@korea.ac.kr


Introduction
In 1977, in doctoral dissertation (thesis advisor, H. Kranzer), Korchinski [1] considered the Riemann problem (RP) for the system: They found that the delta function is supported on a discontinuity, which was motivated by some numerical calculations of the Lax-Friedrich scheme, which was later called a delta shock. For a hyperbolic system of conservation laws, the four-quadrant RP is the initial value problem: U t + f (U) x + g(U) y = 0 (2) with the initial data U(x, y, 0) = U i , i = 1, · · · , 4, where i denotes the ith quadrant. Here, U(x, y, t) = u 1 (x, y, t), · · · , u n (x, y, t) T denotes an n-dimensional vector of conserved quantities, and f and g are nonlinear fluxes. Since Zhang and Zheng's illuminating conjecture [2] on the classification and structure of the solution of four-quadrant RP for 2-D gas dynamics systems in 1990, many studies have been conducted on simplified gas-dynamics-like models.
In 1994, Tan and Zhang [3,4] studied a four-quadrant RP for a 2-D system: In a series of papers, they classified and constructed an analytic solution in the case of initial data involving four contact discontinuities (4J) [3] and in the case of the initial data involving rarefaction waves [4]. They found the delta shock in this model independently of Korchinski, and Tan et al. [5] established the concept of a delta shock. In addition, Pang et al. [6,7], and Shen [8] considered three constant initial data for the system (3).
Under the assumption that ρ = const. ≡ 1, Zhang et al. [9] considered a four-quadrant RP for pressure-gradient equations (4) of the compressible Euler system: where p is the pressure, (u, v) is the velocity, and E = (u 2 + v 2 )/2 + p is the energy. Shen and Sun [10] considered the system (4) with the initial data of three constants in three fan domains.
Cheng et al. [12] considered the RP for the zero-pressure gas dynamics model (5) with three constant states, and Mach reflection-like configurations appear in some of the solutions.
The 2-D RP for Chaplygin gas dynamics with three and four constant states were investigated by Wang et al. [13] and Chen and Qu [14], respectively. Shearer [15] studied the RP for 2 × 2 systems of nonstrictly hyperbolic conservation laws with quadratic nonlinearities that are nondegenerate. In addition, Andrianov and Warnecke [16] considered the RP for compressible duct flows.
Most of the RPs cited above were applied under the following assumption [17]: (H) Outside a neighborhood of the origin, each jump of the initial data projects exactly one plane elementary wave.
This is a restricted assumption for systems because an n × n system can generally develop n waves at each initial discontinuity. Most of the RPs citations below were done without the above assumption (H).
In a series of papers [18,19], Hwang and Lindquist solved the 2-D RP of the system (6), which is a generalization of the 1-D Keyfitz-Kranzer-Issacson-Temple model [20][21][22]: where Under the fundamental assumption of u = v, Sun [23] constructed six topologically distinct solutions for a nonstrictly hyperbolic system (8) : Shen et al. [24] constructed ten solutions for the system (9): This equation can also be considered to be a simplified gas-dynamics-like model because it can be derived from 2-D isentropic Euler equations [25]. Because this is an isotropic case ( f = g in Equation (2)), they considered interactions in the η > ξ plane by applying the transformation and symmetry. Hwang et al. [26] considered a 2-D RP with three constant initial data for the system (9). They classified and constructed 12 solutions. In this paper, our interest is to classify and construct the analytic solution for a 2 × 2 hyperbolic system (9) in the case of initial four-quadrant data without the assumption (H). We consider Riemann's initial data involving delta shock. The numerical solution is also computed for comparison with the analytical solution.
We apply the 2-D direct construction method within the entire plane. This approach has certain advantages because the wave interactions in the full plane can be seen, and the constructed analytic solutions can be compared with numerical solutions. The results show extremely interesting structures of wave interactions of the RP, and the numerical solution is remarkably coincident with the constructed analytic solution. Because the theory gives little insight for the qualitative behavior of wave interactions, we need to construct the solution for each system. Furthermore, since general theory does not exist for multidimensional system, 2-D RP for systems must be investigated on a case-by-case basis.
In Section 2, preliminaries, including elementary waves, are presented. In Section 3, the initial data involving the delta shock are formally classified as 52 cases, which results in 14 solutions. We construct analytic and numerical solutions for each case.

Preliminaries
The system (9) is changed to a self-similar form through ξ = x/t, η = y/t. The system (10) has two eigenvalues and the corresponding right eigenvectors: Then, the λ 1 field is linearly degenerated and the λ 2 field is genuinely nonlinear if η = ξ and u = 0. From the initial discontinuity between two sides (u l , ρ l ) and (u r , ρ r ) in the counterclockwise direction, we use the notation R lr , J lr , S lr , S δ lr for the rarefaction wave, contact discontinuity, shock, and delta shock, respectively.
From the Rankine-Hugoniot condition, the contact discontinuity satisfies which shows that it is directed to the singular point (u l , u l ) = (u r , u r ). Here, J lr (ξ) can be expressed as Again, from the jump condition, the shock satisfies which shows that it directs to the singular point (u l + u r , u l + u r ). The entropy condition is defined as three characteristic "incoming" lines and the remaining "outgoing" line. The shock S lr (ξ) can be expressed as follows: S lr (ξ) : ξ = u l + u r , ρ l u l = ρ r u r , 0 < u l < u r or u l < u r < 0 for η > ξ, 0 < u r < u l or u r < u l < 0 for η < ξ .
Again, from the generalized Rankine-Hugoniot condition, the delta shock is directed to the singular point (u l + u r , u l + u r ). The entropy condition of delta shock implies that all characteristics on both sides of the delta shock are "non-outgoing" [24]. The delta shock S δ lr (ξ) can be expressed as follows: The delta shock is described in detail in [3,5,24,27,28]. However, because the second characteristic line satisfies which shows that it directs (2u, 2u), R lr (ξ) can be expressed as follows: Here, (13), (15), (16), and (18) represent the waves that are parallel to the η-axis. The waves parallel to the ξ-axis can be described in a similar manner.

Construction of the Solution
We consider a four-quadrant RP for system (9). From the entropy condition, all four characteristics are incoming from both sides of the delta shock. This means that 0 is between the two states. Thus, we must consider the order of u 1 , u 2 , u 3 , u 4 , and 0 because of delta shock, which brings a total of 5! = 120 cases. The cases including a delta shock are reduced to 52 cases, which resulted in 14 topologically distinct solutions.
Because we remove the assumption (H), there is one or two waves at infinity for each discontinuity. Consider the values of u on either side of the initial discontinuity. If they have the same sign, then we have two waves: a contact shock (JS) or contact rarefaction (JR). If they have different signs, then we have only one wave: S δ or R.
Because one of the fields of the system (9) is linearly degenerate, contact discontinuity appears as a solution of the RP. For the numerical solution, to reduce the numerical dissipations of the contact discontinuity, the flux functions of the semi-discrete central upwind scheme are modified. Further details can be found in [26,29,30]. In this study, the computational domain is [−4, 4] × [−4, 4] and t = 0.2, ρ i = 0.77 for i = 1, · · · , 4. We use 1200 × 1200 cells, and the CFL is 0.05.

Two Delta Shocks
For the classification of waves at the initial discontinuities, we count the exterior waves that come from the positive η-axis at first, and then at the axes in the counterclockwise direction. In the classification of the initial data, 21034 and 41032 indicate that

No Shock Wave
The upper and lower sides of the line η = ξ form a symmetrical structure in each of the following cases From the initial discontinuity at the positive η-axis, the contact J 1a and the rarefaction R a2 are developed, and a new state (u a , ρ a ) satisfies u a = u 1 and ρ a u a = ρ 2 u 2 . Here, J 1a is directed to the singular point (u 1 , u 1 ), and the rarefaction R a2 is directed to the singular point (2u, 2u) for u 2 ≤ u ≤ u 1 . From the initial discontinuity at the negative ξ-axis, the exterior wave connecting states (u 2 , ρ 2 ) and (u 3 , ρ 3 ) is S δ 23 parallel to the negative ξ-axis. In addition, S δ 23 is directed to the singular point (u 2 + u 3 , u 2 + u 3 ). From the initial discontinuity at the negative η-axis, J 3c and R c4 are developed, and a new state (u c , ρ c ) satisfies u c = u 3 and ρ c u c = ρ 4 u 4 . Moreover, J 3c is directed to the singular point (u 3 , u 3 ), and the rarefaction R c4 is directed to (2u, 2u) for u 3 ≤ u ≤ u 4 . From the initial discontinuity at the positive ξ-axis, the exterior wave connecting states (u 4 , ρ 4 ) and (u 1 , ρ 1 ) is S δ 41 parallel to the positive ξ-axis. Finally, S δ 41 is directed to the singular point (u 4 + u 1 , u 4 + u 1 ).
The straight delta shock S δ 23 completely penetrates the rarefaction wave R a2 at point , and then the curved delta shock η = η(ξ) from A to B(2u 1 , and we obtain This straight delta shock S δ a3 meets the contact discontinuity J 1a at point C(u 1 , ) and continues to the singular point D(u 1 + u 3 , u 1 + u 3 ), and has the form By contrast, the delta shock S δ 41 penetrates the whole rarefaction wave R c4 , and the straight delta shock S δ c1 from the point E(2u 3 , From the initial discontinuity, a delta shock is formed at the negative ξ-axis and negative η-axis, and contact rarefaction is formed at the positive η-axis and positive ξ-axis. The delta shock S δ 23 penetrates the entire rarefaction wave R a2 . The straight delta shock S δ a3 from point A(2u 1 , ) meets the contact discontinuity J 1a at point C(u 1 , and continues to the singular point B(u 1 + u 3 , u 1 + u 3 ).
By contrast, the delta shock S δ 34 completely penetrates the rarefaction wave R 4d .
The straight delta shock S δ 3d from the point D( In this case, only one wave is formed at each initial discontinuity. From the initial discontinuity at the positive η-axis, the exterior wave connecting states (u 1 , ρ 1 ) and (u 2 , ρ 2 ) is the rarefaction R 12 parallel to the positive η-axis. The rarefaction R 12 is directed to (2u, 2u) for u 2 ≤ u ≤ u 1 . From the initial discontinuity at the negative ξ-axis, the exterior wave connecting states (u 2 , ρ 2 ) and (u 3 , ρ 3 ) is S δ 23 , which is directed to the singular point (u 2 + u 3 , u 2 + u 3 ). From the initial discontinuity at the negative η-axis, the exterior wave connecting states (u 3 , ρ 3 ) and (u 4 , ρ 4 ) is S δ34 , which is directed to the singular point (u 3 + u 4 , u 3 + u 4 ). From the initial discontinuity at the positive ξ-axis, the exterior wave connecting states (u 4 , ρ 4 ) and (u 1 , ρ 1 ) is the rarefaction R 41 , which is directed to (2u, 2u) for u 4 ≤ u ≤ u 1 .
The straight delta shock S δ 23 meets the rarefaction wave R a2 at point A(2u 2 , u 1 + u 2 ), and the curved delta shock η = η(ξ) from A to B(0, and we obtain Simultaneously, a new shock and a new contact discontinuity occur at B. This is an extremely interesting phenomenon in which the delta shock is switched into shock and contact discontinuity. This new shock satisfies the equations in (22) in 0 ≤ u ≤ u 3 . Similarly, this curved shock has the same form as (23), and continues to C(2u 3 , 2u 3 ). In addition, J e3 formed at point B ends at the singular point D(u 3 , u 3 ) and satisfies By contrast, S δ 34 meets R 41 at the point E(u 3 + u 4 , 2u 4 ). The curved delta shock continues to point F( , 0), and J 3e and S eu (0 ≤ u ≤ u 3 ) are formed at F. This shock continues to point C, and J 3e ends at the singular point D. This contact discontinuity J 3e meets the contact discontinuity J e3 at point D. Both rarefaction waves R 12 and R 41 meet at the same singular point (2u, 2u) for u 3 ≤ u ≤ u 1 between C and G(2u 1 , 2u 1 ). The solutions are shown in Figure 3. The initial condition is u 1 = 0.56, u 2 = −0.37, u 3 = 0.43, u 4 = −0.15.
As in case 3, only one wave, either a delta shock or rarefaction, is formed at each initial discontinuity. The straight delta shock S δ 12 meets the rarefaction wave R 23 at point A(u 1 + u 2 , 2u 2 ). The curved delta shock continues from A to B(
By contrast, the delta shock S δ 34 intersects with the shock S 4d at point D(u 3 + u 4 , u 4 + u 1 ). Then, the new delta shock S δ 3d from D to B satisfies The straight delta shock S δ 3d meets the contact discontinuity J d1 at point E(

Case 6. JS
From the initial discontinuity, a delta shock is formed at the negative ξ-axis and negative η-axis, and contact shock and contact rarefaction contact are formed at the positive η-axis and positive ξ-axis, respectively. The delta shock S δ 23 meets the shock S a2 at point A(u 1 + u 2 , u 2 + u 3 ). The new delta shock S δ a3 formed at A meets the contact discontinuity J 1a at C(u 1 , u 1 u 2 −u 2 3 u 2 −u 3 ) and continues to the singular point B(u 1 + u 3 , u 1 + u 3 ). By contrast, the delta shock S δ 34 penetrates the entire rarefaction wave R 4d , and the straight delta shock S δ 3d from the point D(
However, the delta shock S δ 41 intersects with the shock S c4 at the point D(u 3 + u 4 , u 4 + u 1 ). The new delta shock S δ c1 formed at D meets the contact discontinuity J 3c at the point From the initial discontinuity, a delta shock is formed at the negative ξ-axis and negative η-axis, and a contact shock is formed at the positive η-axis and positive ξ-axis. The delta shock S δ 23 meets the shock S a2 at point A(u 1 + u 2 , u 2 + u 3 ). The new delta shock S δ a3 formed at A meets the contact discontinuity J 1a at C(u 1 , By contrast, the delta shock S δ 34 intersects with the shock S 4d at point D(u 3 + u 4 , u 4 + u 1 ).
The new delta shock S δ 3d formed at D meets the contact discontinuity J d1 at E(u 3 ,

One Shock Wave
Case 10 : Case 12 : JR + JS + R + S δ (21304), S δ + R + SJ + RJ (41302) R + S δ + JR + JS (20134), SJ + RJ + S δ + R (40132) Case 13 : JR + JS + R + S δ (23104), S δ + R + SJ + RJ (43102) R + S δ + JR + JS (20314), SJ + RJ + S δ + R (40312) Case 10. S δ + R + RJ + SJ (u 1 < u 3 < u 4 < 0 < u 2 ) From the initial discontinuity, a delta shock is formed at the positive η-axis, and a rarefaction is formed at the negative ξ-axis, and the rarefaction-contact and shock-contact are formed at the negative η-axis and positive ξ-axis, respectively. The delta shock S δ 12 meets the rarefaction wave R 23 at point A(u 1 + u 2 , 2u 2 ). Then, the curved delta shock continues toward point B( u 2 1 u 1 −u 2 , 0). A new shock and a new contact discontinuity are formed at B, and this curved shock ends at point C( , 2u 3 ). Then, S e3 continues from point C to D(u 1 + u 3 , u 1 + u 3 ), and J 1e continues from point B to point E(u 1 , u 1 ). By contrast, J c4 intersects with S 4d at point F(u 4 , u 4 + u 1 ) and the new contact discontinuity J ed from F ends at their singular point E, which equals the singular point of J d1 and J 1e . Therefore, the three contact discontinuities meet at the singular point E. The shock S ce (= S 4d ) completely penetrates R 3c and then stops at the singular point D. The solutions are shown in Figure 10. The initial condition is u 1 = −0.73, u 2 = 0.15, u 3 = −0.56, u 4 = −0.25.  Case 11. R + S δ + JS + JR (u 2 < 0 < u 4 < u 3 < u 1 ) From the initial discontinuity, rarefaction is formed at the positive η-axis, and a delta shock is formed at the negative ξ-axis, and contact shock and contact rarefaction are formed at the negative η-axis and positive ξ-axis, respectively. The delta shock S δ 23 meets the rarefaction wave R 12 at point A(2u 2 , u 2 + u 3 ), and the curved delta shock continues toward point B(0, u 2 3 u 3 −u 2 ). Simultaneously, S ue (0 ≤ u ≤ u 3 ) and J e3 occur at B. The curved shock continues from B toward C(2u 3 , 2u 3 ). In addition, J e3 formed at point B ends at the singular point D(u 3 , u 3 ).
By contrast, J 4d meets S c4 at point E(u 3 + u 4 , u 4 ) and J ce at the singular point D. Therefore, three contact discontinuities J e3 , J 3c , and J ce meet at their singular point D. The shock S ed (= S c4 ) meets the rarefaction wave R d1 at point F(u 3 + u 4 , 2u 4 ), and the curved shock continues toward C. Both rarefaction waves R 12 and R d1 meet at (2u, 2u) for u 3 ≤ u ≤ u 1 between C and G(2u 1 , 2u 1 ). The solutions are shown in Figure 11. The initial condition is u 1 = 0.56, u 2 = −0.15, u 3 = 0.43, u 4 = 0.21.
From the initial discontinuity, a delta shock is formed at the positive η-axis and rarefaction is formed at the negative ξ-axis; in addition, shock-contact and rarefactioncontact are formed at the negative η-axis and positive ξ-axis, respectively. The straight delta shock S δ 12 meets R 23 , and then a curved delta shock continues toward A( u 2 1 u 1 −u 2 , 0). The new shock formed at A completely penetrates R 23 and stops at the singular point B(u 1 + u 3 , u 1 + u 3 ). Moreover, J 1e formed at point A ends at the singular point C(u 1 , u 1 ). However, the shock S 3c completely penetrates the rarefaction wave R 4d from D to the point E( u 2 1 +u 2 3 −2u 1 u 4 u 3 −u 4 , 2u 1 ) and it stops at the singular point B. In addition, J c4 completely penetrates R ce (= R 4d ) from F(u 4 , 2u 4 ) to G and satisfies which gives The straight contact discontinuity J ed continues from G( Here, J ed meets J 1e and J d1 at their singular point C. The solutions are shown in Figure 12. The initial condition is u 1 = −0.43, u 2 = 0.15, u 3 = −0.37, u 4 = −0.56.  Case 13. S δ + R + SJ + RJ (u 4 < u 3 < u 1 < 0 < u 2 ) In this case, the exterior waves at the initial discontinuity are exactly the same as those in case 12. The upper structures are also extremely similar to those in the previous case.

Two Shock Waves
Case 14 : JS + JS + R + S δ (12304), S δ + R + SJ + SJ (14302) R + S δ + JS + JS (20143), SJ + SJ + S δ + R (40123) Case 14. JS + JS + R + S δ (u 1 < u 2 < u 3 < 0 < u 4 ) From the initial discontinuity, a contact shock is formed at the positive η-axis and negative ξ-axis, and rarefaction and delta shock are formed at the negative η-axis and positive ξ-axis, respectively. In addition, J 2b intersects with S a2 at point P(u 1 + u 2 , u 2 ) and the new contact discontinuity J ae occurs from P to A(u 1 , u 1 ). The shock S a2 (= S eb ) continues from P without changing the direction because u e = u a and u b = u 2 . This shock S eb meets the shock S b3 at the point Q(u 1 + u 2 , u 2 + u 3 ), and the new shock S e3 is formed at Q and ends at the singular point H(u 1 + u 3 , u 1 + u 3 ). By contrast, S δ 41 meets R 34 at the point R(2u 4 , u 4 + u 1 ), and the curved delta shock is then from R to S(0, u 2 1 u 1 −u 4 ). Simultaneously, J e1 and S ue (u 3 ≤ u ≤ 0) are formed at S. The curved shock from S to T and the straight shock continues from the point T(2u 3 , ) and meets S e3 at the singular point H(u 1 + u 3 , u 1 + u 3 ). However, the new contact discontinuity J e1 is from S to A and meets two contact discontinuities J 1a and J ae at their singular point A(u 1 , u 1 ). The solutions are shown in Figure 14. The initial condition is u 1 = −0.56, u 2 = −0.43, u 3 = −0.37, u 4 = 0.15.