Delta Shock Wave Research Articles

Non‐self‐similar solutions containing delta shock waves for nonhomogeneous zero‐pressure gas dynamics with energy conservation law by the viscosity method

This article considers the zero‐pressure gas dynamics governed by the conservation laws of mass, momentum, and energy with time‐dependent source terms. By adopting the vanishing viscosity method, we propose a viscous system to prove the stability of non‐self‐similar Riemann solutions including the delta shock wave and vacuum.

Concentration and cavitation in the Riemann solution for the relativistic Euler equations with non-ideal isothermal dusty gas

The phenomena of concentration and cavitation in the Riemann solutions of the relativistic Euler equations with non-ideal isothermal dusty gas are considered. The Riemann problem is solved first, and the solutions consisting of rarefaction waves and shock waves are obtained. Then, by the vanishing pressure approach, we investigate the limiting behaviors of Riemann solutions and observe the concentration and cavitation phenomena. It is rigorously proved that as the dusty gas pressure vanishes with double parameters, the solution containing two shock waves and two rarefaction waves respectively converges to the delta-shock wave and vacuum state of the pressureless relativistic Euler equations.

Delta shock wave for ternary nonlinear chromatography equations as self-similar viscosity limit

In this paper, we investigate the existence and stability of the delta shock wave in ternary chromatography equations by the self-similar viscosity vanishing approach. Considering the appropriate initial values, we prove the existence of the self-similar solution for the corresponding Riemann problem of the ternary chromatography viscous equations. Furthermore, we rigorously demonstrate that the delta shock wave is the weak star limit of the self-similar solution as viscosity tends to disappear. The result implies that the structure of the delta shock wave is stable under the self-similar viscosity perturbation, which guarantees that the delta shock wave is a unique entropy solution. In addition, we present numerical simulations in agreement with the theoretical analysis.

Concentration and cavitation phenomena of Riemann solutions for the generalized Chaplygin gas equations under the flux approximation

In this paper, we investigate the concentration and cavitation phenomena of Riemann solutions for the generalized Chaplygin gas equations in the presence of flux approximation. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums (or constant density states), respectively. The main objective of this paper is to rigorously investigate the formation of delta shock waves and constant density states and observe the concentration and cavitation phenomena. First, the Riemann problem for the generalized Chaplygin gas equations under the flux approximation is solved constructively. Although the system is strictly hyperbolic and its two characteristic fields are genuinely nonlinear, the delta shock wave arises in Riemann solutions. The formation of mechanism for delta shock wave is analyzed, that is, the 1-shock wave curve and the 2-shock wave curve do not intersect each other in the phase plane. Second, it is rigorously proved that, as the pressure vanishes, the Riemann solutions for the generalized Chaplygin gas equations under the flux approximation tend to the two kinds of Riemann solutions to the transport equations in zero-pressure flow under the flux approximation, which include a delta shock wave formed by a weighted δ-measure and a constant density state.

Piston problem for the pressureless hydrodynamic traffic flow model

The hydrodynamic model can be used to describe traffic problems in transport. When the speed of the first car is less than the speed behind it, it leads to traffic jams. When the first car's speed is faster than the cars behind it, it leads to traffic evacuation. If we consider the first car to be a piston, then the speed of the piston will cause traffic jams and traffic evacuation. In this paper, we study the piston problem for the hydrodynamic model. The formation and propagation of shock wave, rarefaction wave, delta-shock wave, and vacuum can describe the phenomena of traffic jams, traffic evacuation, severe traffic jams, and traffic evacuation with traffic volume of zero, respectively. Therefore, for different traffic phenomena, we prove the existence of shock solution, rarefaction solution, delta shock solution, and vacuum solution. In addition, we perform some representative numerical simulations.

The Riemann problem for the Chaplygin gas dynamics with a single-point heating source

This paper is devoted to the Riemann solutions to the Chaplygin gas dynamics with a single-point heating source. To deal with Dirac measure source term, we transform this model into a system of conservation laws and then present a definition of solution to this system of conservation laws. Based on this definition, we establish, respectively, the condition for the occurrence of stationary contact discontinuity solution and delta standing wave solution. With the help of these results, by analyzing the possible combination of various waves, ten kinds of exact solutions and the criteria for the emergence of each solution are established. It is observed that a delta standing wave solution arises for certain initial values, where two components of this solution contain the Dirac measure. It is also noticed that a stationary contact discontinuity followed by a delta shock wave emerges in some solutions. Moreover, these solutions can be used to test the validation of numerical algorithm for a system of conservation laws with a singular source term.

The concentration and cavitation in the Riemann solution for non-homogeneous logrithmic equation of state with magnetic field

In the present study, the phenomena of concentration and cavitation in the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field is analyzed. Firstly, we introduced new state variable for the velocity to modify the non-conservative system into conservative system and solved the Riemann problem for modified system constructively. Further, the Riemann solutions for the transport equations is investigated as pressure and magnetic field vanish. It is proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two shock waves converges to the delta shock wave solution of the transport equations as pressure and magnetic field vanish. It is also proved that the Riemann solution for the non-homogeneous hyperbolic system with logarithmic equation of state and magnetic field having two rarefaction waves converges to the contact discontinuity solution of the transport equations as pressure and magnetic field vanish.

The vanishing pressure limits of Riemann solutions for the Aw-Rascle hydrodynamic traffic flow model with the logarithmic equation of state

Two kinds of Riemann solutions for the Aw-Rascle hydrodynamic traffic flow model with the logarithmic equation of state are explicitly obtained by using the combination between 1-rarefaction or 1-shock wave along with 2-contact discontinuity. The formation of vacuum state and delta shock wave is identified and analyzed when the perturbation parameter in the pressure term drops to zero, where the intrinsic cavitation and concentration phenomena are surveyed and explored concretely. Additionally, several numerical results displaying the formation process of vacuum state and delta shock wave are also presented by taking three different perturbation parameters for comparison.

The composite wave in the Riemann solutions for macroscopic production model

In this paper, the Riemann solutions for the macroscopic production model with chaplygin gas under some special initial data are constructively obtained in the fully explicit form. An interesting composite wave RδJ is observed, it is formed by a rarefaction wave R and a left-contact delta discontinuity δJ attached to the wavefront of the rarefaction wave. Furthermore, this delta discontinuity gradually absorbs the rarefaction wave and eventually forms a delta shock wave under some suitable initial condition. In addition, we give some typical numerical results that coincide well with the theoretical analysis.

Multiplication of distributions and singular waves in several physical models

We study of a Riemann problem for a one-dimensional 2×2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\hspace{1.111pt}{\ imes }\\hspace{1.111pt}2$$\\end{document} system of conservation laws, which includes some particular cases of known physical systems, such as the pressureless gas dynamics system, Euler equations for isentropic flow, the Brio system, and the shallow water system. The main results of this study reveal the emergence of shock waves and delta shock waves as explicit solutions. We use the concept of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-solution defined in the setting of a product of distributions not defined by approximation. Notably, this study does not assume any classical results about conservation laws, presenting a simpler and more general framework for constructing singular solutions to other equations or systems. Additionally, this paper includes several comments on the four physical systems mentioned, along with formulas for multiplying distributions and evaluating the compositions of a function with a distribution.

The limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force

In this paper, we investigate the limiting behavior of Riemann solutions to the Euler equations of compressible fluid flow for modified Chaplygin gas with the body force as the two parameters tend to zero. The formation of delta shock waves and the vacuum states is identified and analyzed during the process of vanishing pressure in the Riemann solutions. The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, our main objective is to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force is solved. Second, we rigorously confirm that, as the pressure vanishes, any two shock Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a δ-shock solution to the pressureless gas dynamics model with a body force, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the δ-shock; any two-rarefaction-wave Riemann solution to the Euler equations of compressible fluid flow for the modified Chaplygin gas with the body force tends to a solution consisting of four contact discontinuities together with vacuum states with three different virtual velocities in the limiting situation.

The free piston problem for pressureless Euler equations under the gravity

In this paper, the exact solutions of the free piston problem for the one-dimensional pressureless Euler equations under gravity are constructed in fully explicit forms. It is found that delta shock wave and vacuum state are contained in the solutions with appropriate initial condition. Furthermore, the trajectory function of the piston can be obtained in the combination of the generalized Rankine-Hugoniot conditions and Newton's second law, and it can be uniquely determined by the generalized entropy condition. We observe and analyze carefully some interesting nonlinear phenomena. For instance, delta shock wave absorbs the vacuum state and forms a new delta shock wave. In addition, some representative numerical simulations are presented to confirm the correctness of theoretical analysis.

Flux approximation to the Euler equations of compressible fluid flow for extended Chaplygin gas

The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, we are concerned with the Euler equations of compressible fluid flow when the state equation is governed by the extended Chaplygin gas, an important candidate for describing dark matter and dark energy. Our main objective is to apply the flux-approximation method to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the compressible fluid flow with n + 2 parameters including flux and pressure is solved. Then, two kinds of flux approximation are discussed, that is, both the flux perturbation and pressure tend to zero, or only the flux perturbation vanishes while the extended Chaplygin gas pressure partly tends to the Chaplygin gas pressure. The results indicate that different manners of flux approximation have their respective effects on the formation of delta shock waves. Finally, several numerical results are presented to confirm the theoretical analysis.

Solution of hyperbolic system with magnetic field

This paper is concerned with the solution of the Riemann problem (RP) for the hyperbolic system with the logarithmic equation of state and magnetic field. The formation of vacuum states and delta shock wave as magnetic field and pressure vanish have been discussed. First, the Riemann problem for the magnetogasdynamic system is solved. Further, we have constructed the solutions for the pressureless and vanishing magnetic field system (i.e. transport equations). In the context of vacuum states and delta shocks, it is shown that the solution of RP consisting two shocks converges to the delta shock wave (δ-shock) solution of transport equations, and the Riemann solutions consisting two rarefaction waves converge to the vacuum state solution (i.e. intermediate state between two-contact discontinuity solution of transport equations). Hence, it is proved that the solutions of system with logarithmic equation of state and magnetic field converge to the solution of corresponding system as magnetic field and pressure vanish, which shows that the solution of the RP for the hyperbolic system with the logarithmic equation of state and magnetic field is stable.

The perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction

In this paper, we are concerned with the Riemann problem and the perturbed Riemann problem for the Chaplygin pressure Aw–Rascle model with Coulomb-like friction, which can also be seen as the nonsymmetric Keyfitz–Kranzer system with Chaplygin pressure and Coulomb-like friction. For the Riemann problem, we show that it explicitly exhibits two kinds of different structures and the delta shock wave appears in some certain situations. The generalized Rankine–Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly. Unlike the homogeneous case, it is shown that the Coulomb-like friction term makes contact discontinuities and the delta shock wave bend into parabolic shapes and the Riemann solutions are not self-similar anymore. For the perturbed Riemann problem with delta initial data, not only the delta shock wave but also the delta contact discontinuity are found in solutions and the friction term makes them bent. Under the generalized Rankine–Hugoniot conditions and the entropy condition, by taking variable substitution, we constructively obtain the global existence of generalized solutions which explicitly exhibit four kinds of different structures. The results in this paper yield a way of studying the wave interaction involving the delta shock wave for conservation laws with source terms and will give us some insights into the research on the Chaplygin pressure Aw–Rascle model, the pressureless Euler equations and the Chaplygin Euler equations with various kinds of source terms.

Asymptotic behavior of Riemann solutions to a hyperbolic system arising from a traffic flow model

This paper is mainly concerned with the Riemann problem for a hyperbolic system arising from a traffic flow model, whose Riemann solutions are obtained constructively in terms of the fine analysis of elementary waves. Especially, the Dirac delta shock wave is involved in the Riemann solution at some specially designated initial data. Then, the limiting behavior of Riemann solutions is investigated carefully from the current system to the pressureless Euler system by sending the perturbed parameter tend to zero, in which more complicated nonlinear phenomena of cavitation and concentration than before can be observed and analyzed during this limiting process. Finally, several representative numerical examples are offered to confirm our theoretical results.

Limits of solutions to the Aw-Rascle traffic flow model with generalized Chaplygin gas by flux approximation

The Riemann problem for the Aw-Rascle (AR) traffic flow model with a double parameter perturbation containing flux and generalized Chaplygin gas is first solved. Then, we show that the delta-shock solution of the perturbed AR model converges to that of the original AR model as the flux perturbation vanishes alone. Particularly, it is proved that as the flux perturbation and pressure decrease, the classical solution of the perturbed system involving a shock wave and a contact discontinuity will first converge to a critical delta shock wave of the perturbed system itself and only later to the delta-shock solution of the pressureless gas dynamics (PGD) model. This formation mechanism is interesting and innovative in the study of the AR model. By contrast, any solution containing a rarefaction wave and a contact discontinuity tends to a two-contact-discontinuity solution of the PGD model, and the nonvacuum intermediate state in between tends to a vacuum state. Finally, some representatively numerical results consistent with the theoretical analysis are presented.

Stability of Riemann solutions to a class of non-strictly hyperbolic systems of conservation laws

This paper is concerned with Riemann problem for a class of non-strictly hyperbolic systems of conservation laws. The two kinds of Riemann solutions including vacuum and delta shock wave are constructed. Under the generalized Rankine-Hugoniot relation and entropy condition, we establish the existence and uniqueness of delta shock wave solutions. Furthermore, by studying the interactions among of the delta shock wave and vacuum as well as contact discontinuity, the Riemann solutions with four kinds of different structures are obtained. Additionally, the stability of the Riemann solutions is obtained under certain perturbation of the initial data.

The Riemann problem for a traffic flow model

A traffic flow model describing the formation and dynamics of traffic jams was introduced by Berthelin et al. [“A model for the formation and evolution of traffic jams,” Arch. Ration. Mech. Anal. 187, 185–220 (2008)], which consists of a pressureless gas dynamics system under a maximal constraint on the density and can be derived from the Aw–Rascle model under the constraint condition ρ≤ρ* by letting the traffic pressure vanish. In this paper, we give up this constraint condition and consider the following form: {ρt+(ρu)x=0,(ρu+εp(ρ))t+(ρu2+εup(ρ))x=0,in which p(ρ)=−1ρ. The Riemann problem for the above traffic flow model is constructively solved. The delta shock wave arises in the Riemann solutions, although the system is strictly hyperbolic, its first eigenvalue is genuinely nonlinear, and the second eigenvalue is linearly degenerate. Furthermore, we clarify the generalized Rankine–Hugoniot relations and δ-entropy condition. The position, strength, and propagation speed of the delta shock wave are obtained from the generalized Rankine–Hugoniot conditions. The delta shock may be useful for the description of the serious traffic jam. More importantly, it is proved that the limits of the Riemann solutions of the above traffic flow model are exactly those of the pressureless gas dynamics system with the same Riemann initial data as the traffic pressure vanishes.

Construction of solutions of a two‐dimensional Riemann problem for a thin film model of a perfectly soluble antisurfactant solution

This article is concerned with formulation of three‐dimensional thin film model for an antisurfactant solution and hence constructing unique global solution for a two‐dimensional Riemann problem for the corresponding reduced hyperbolic form. We develop six geometrically different structures of the solution using generalized characteristic analysis method while relaxing the restriction that only one planar elementary wave is developed at the interface of each initial discontinuity. We analyze the interactions of classical and nonclassical waves in detail to construct the global solution of the corresponding 2‐D Riemann problem. Further, we provide the expressions for strength, location, and propagation speed of delta shock wave at each interaction point. Moreover, we compare these solutions with the solutions of a one‐dimensional rotated initial value problem and prove that our solutions are globally unique.