Riemann problems and delta-shock solutions for a Keyfitz-Kranzer system with a forcing term
Introduction
In this paper, we study the Riemann solutions for a nonsymmetric Keyfitz-Kranzer system with source term: We consider two types of source term. A first type is given by , where α is a nonzero constant friction coefficient. In this case, the system (1) is called Coulomb-like friction model. The Riemann problem to the system (1) with was studied in [17] and the interaction of delta-shock waves in [16]. A second case that we consider in this work is when and in this case the system is called linear damping model. To study the Riemann problem to the system (1), we consider the following initial data for arbitrary constant states with . We show the existence of an intricate delta-shock wave solution and its generalized Rankine-Hugoniot condition resulting from the analysis, which was used in a comprehensive study of Riemann problems and delta-shock solutions for the nonsymmetric Keyfitz-Kranzer system with forcing term (1).
We recall that delta-shock is a type of nonclassical wave solution in which at least one state variable may develop a Dirac delta measure [5], [29], [31], [34], [46], [48]. Actually on physical grounds, delta shock solutions typically display concentration occurrence in a complex system. From the mathematical viewpoint, delta-shocks are distinguished delta type-functions that emerges in the state variables and initial data for some classes of equations. We mention the work of H. Yang and collaborators [13], [35], [52], [53], [54] for recent achievements of delta-shock wave structures and its applications in systems of hyperbolic conservation laws. In particular, in Yang [53] there is given a detailed review about the generality and applications on the delta shock waves, starting from the PhD thesis of Korchinski 1977 [32] and then presenting and discussing several pertinent aspects on the development of theory of the delta-shock waves (e.g., [6], [7], [10], [13], [26], [35], [36], [41], [44], [52], [53], [54]) covering interesting results from 1989 until the paper of H. Yang and collaborators [35] in 2020, where Riemann solutions describing the formation processes of delta-shocks and vacuum states are clarified for an intricate pressureless magnetohydrodynamics system. Over these four decades, the occurrence of delta-shocks has emerged in many areas such as chromatography [53], magnetohydrodynamics [35], [51], traffic flow [36], fluid dynamics [30], and possibly in flow in porous media [6]; see also [8], [39] and references cited therein for other type of nonclassical wave appearing in the modeling phenomena of flow and hyperbolic-transport in porous media. Compared with the delta-shocks, we also mention the papers [26], [41], [44] for -shocks and [12] for noncompressible δ-waves. Moreover, delta-shocks also appear in many other situation as such concentration and cavitation (vacuum state) in the vanishing pressure limit of solutions related to the Euler equations for nonisentropic fluids [10], [52].
In [13], the authors discussed this topic by investigating the partly vanishing pressure limits of solutions to a nonsymmetric Keyfitz-Kranzer system of conservation laws with generalized and modified Chaplygin gas. The existence of solutions of the Cauchy problem for the system (3) with inhomogeneous source terms, , f being a nonnegative convex function and P satisfying a suitable condition, was shown in [18]. It is worth mentioning that there are very few developments of Riemann problems with delta-shock waves. For instance, in [21] the authors considered Riemann problems to the inhomogeneous modified Chaplygin gas equations as the pressure vanishes. In [55], the Riemann problems for a class of coupled hyperbolic systems of conservation laws with a source term are studied, which the Riemann solutions include two kinds: delta-shock solutions and vacuum solutions. In [15], the Riemann problem for a hyperbolic system with linear damping was studied using a special time-dependent viscosity to obtain by limiting viscosity approach the classical Riemann solutions and delta-shock.
Here we study Riemann problems and delta-shock solutions for a Keyfitz-Kranzer system with a forcing term, Eq. (1), with source term given by Coulomb-like friction term or by linear damping. As in [17], we consider a more general function f, such that , with for all , and and we obtain the classical Riemann solutions to the problem (1) with initial data (2). The homogeneous case of the system (1) is a particular case of which has been associated with various physical applications, depending on the function ϕ. For example, when and the system (3) was first derived as a model for the elastic string by Keyfitz and Kranzer [28]. So, the system (3) is called Keyfitz-Kranzer. In [25], the system (3) models the polymer flooding of an oil reservoir. Another important study on the system (3) is due to Blake Temple [49] where the author considered the case when is not a convex function (in this case, the system describes how the addition of a polymer affects the flow of water and oil in a reservoir) and proved the existence of a global weak solution to the Cauchy problem. Indeed, we notice when the system (3) was studied by Aw and Rascle [4] as a macroscopic model for traffic flow where ρ and u are, respectively, the density and the velocity of cars on a roadway, and P is a smooth and strictly increasing function satisfying .
Lu [37] showed the existence of a global weak solution of the Cauchy problem for (3) where , f is a non negative convex function and P satisfies the following condition: and .
In [17], the authors solved the Riemann problem of the system (3) with , f been an arbitrary strictly increasing function and for . The pressure function given by is called Chaplygin pressure. This model for the pressure was introduced by Chaplygin [9], as a suitable mathematical approximation for calculating the lifting force on a wing of an airplane in aerodynamics. Observe that is the prototype function satisfying for and it is not possible to solve the Riemann problem to (3) using only classical waves. The results in [17] include delta-shock solutions. So, when , and , the system (3) is the homogeneous case of the system (1) (with ) and the Riemann solution for this case has as solution a 1-contact discontinuity followed by a 2-contact discontinuity where and , if . When , the solution has a form of a delta-shock wave, Here, when , we have and . When , we have
and . Here, the symbol for any function evaluated in the variable A on the left and on the right of a discontinuity curve. The existence and uniqueness of solution for the Riemann problem for the homogeneous case corresponding to the system (1) with initial data (2) is well studied in [17] (and for the particular case in [11]). A complete study of interaction of delta-shock waves was shown by the authors of this paper in [16], which includes several numerical results evidencing delta-shock wave solutions and its interactions with contact discontinuities. Also, it is clear that the homogeneous system corresponding to the system (1) is invariant under uniform stretching of coordinates: with β a constant, hence it admits self-similar solutions defined on the space-time plane and constant along straight-line rays emanating from the origin. But, the structure of solutions for the Riemann problem to the inhomogeneous system (1) is more complicated since there is no self-similar solution in the form of due to the inhomogeneity. The Coulomb-like friction term αρ was introduced by Savage and Hutter [43] to describe the granular flow behavior. Some results about of the Riemann problem for inhomogeneous systems with the Coulomb-like friction term can be found in [40], [45], [56]. On the other hand, in [14] it was studied the Cauchy problem for a system of conservation laws with a frictional damping, and in [38] it was studied the interaction of elementary waves for compressible Euler equations with frictional damping. The reader can find related works in [22], [23], [24], [27]. In our work, we present two types of delta-shock wave solutions, global for Coulomb-like frictional term and local for linear damping. A remarkable behavior for linear damping problem is, under appropriated initial data, the existence of the delta-shock wave solution for a finite time , where , , and the initial data of Riemann type (2). In this case, the delta-shock solution exists for , and for larger times the delta-shock disappears and the solution is composed by contact discontinuities for .
Finally, to confirm the theoretical analysis, we simulate the Riemann solutions and the formation process of delta-shock waves by using a fully-discrete Lagrangian-Eulerian scheme [1], [2], [3], recently introduced in the literature. A key hallmark of the Lagrangian-Eulerian method is the dynamic tracking forward of the no-flow region (per time step). Indeed, another advantage of our proposed Lagrangian-Eulerian scheme is simplicity in the implementation: no (local) Riemann problem is solved and hence time-consuming field-by-field type decompositions are avoided for tracing the direction of the wind (in particular in the case of systems). This is a considerable improvement compared to the classical backward tracking over time of the characteristic curves over each time step interval, which is based on the strong form of the problem. The remainder of this paper is organized as follows.
In Section 2, we present the solution of the classical Riemann problem for the nonsymmetric system of Keyfitz-Kranzer type (1) with source term and . In both cases, Coulomb-like frictional term or linear damping, the initial data (2) satisfies the condition and we obtained a shock wave solution that satisfies the Lax shock condition. In section 3, we study the case when the initial data (2) satisfies the condition in which we have a delta-shock wave solution. For Coulomb-like frictional term the delta-shock wave solution is global while for linear damping the delta-shock wave is local. In Section 4, we briefly present the Lagrangian-Eulerian scheme and a numerical study for a valuable insight beyond its practical applications linked to the Coulomb-like friction term into the Keyfitz-Kranzer model. We also show an interesting numerical example (related to the linear damping case) where the cutoff time in which a delta-shock singular solution ceases to exist is observed (see from Fig. 8 to 10). Finally, our conclusions are given in Section 5.
Section snippets
Case I: Coulomb-like friction term
We study in this section the system (1) with source term . Motivated by Faccanoni and Mangeney [20] for the shallow water equations and also the recently paper due Zhang and Zhang [56], we introduce a new state variable to perform the transformation This type of transformation for variables is an effective approach to study the balance laws with the external force terms like (1). Under the transformation (4), the system (1) and the initial data (2) reduce to
Delta-shock solution
From the previous analysis, we can see that the classical solution does not exist for any initial Riemann data. In Fig. 1.a, we shall denote the common boundary between regions IV and V by . The reason for this notation is that for right states in region V we cannot connect and by the classical shock waves and . Thus, it is for right states in region V that we must use delta-shock waves to solve the Riemman problem for the system (1) with source term. Notice that
Riemann and delta-shock wave solutions: a numerical study
We discuss numerical experiments exhibiting the formation process of delta-shocks and Riemann solutions for the system (1), verifying theory in this work.
Concluding and remarks
In this work, we study the Riemann problem for a nonsymmetric system of Keyfitz-Kranzer type (1) with source term with initial data (2). When the source term is called Coulomb-like friction term and when is called linear damping. For both cases, Coulomb-like friction term or linear damping, if the initial data (2) satisfies the condition the shock wave solution is obtained. Moreover, the shock solution satisfies the Lax shock condition (which is an entropy
Acknowledgments
E. Abreu thanks research grants as well as thanks to all the support given by the Brazilian funding agencies FAPESP 2019/20991-8 (São Paulo), CNPq 306385/2019-8 (National) and Petrobras 2015/00398-0. R. De la cruz gratefully acknowledges the financial support of São Paulo Research Foundation (FAPESP) Grant 2016/19502-4. W. Lambert thanks the Visiting Researcher Grant Supported by São Paulo Research Foundation (FAPESP) Grant 2019/20991-8. E. Abreu also gratefully acknowledge the financial
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