Pressureless Gas Dynamics Research Articles

The singular limits of the Riemann solutions as pressure vanishes for a reduced two-phase mixtures model with non-isentropic gas state

We study the cavitation and concentration phenomena of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic gas state in vanishing pressure limit. We solve the Riemann problem by constructing the regions in (p, u, s) coordinate system. Then we obtain the limiting behaviors of the Riemann solutions and the formation of δ-shock waves and vacuum as pressure vanishes. We conclude that, as pressure vanishes, the limit of Riemann solutions is the Riemann solutions of the reduced 2-dimensional pressureless gas dynamics model. Finally, we present numerical simulations which are consistent with our theoretical analysis.

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.

Limiting behaviour of the Riemann solution to a macroscopic production model with van der Waals equation of state

In our present work, we analyze the limiting behaviour of solution to the Riemann problem for a macroscopic production model with van der Waals equation of state. We construct the solution to the Riemann problem of the governing system which consists of only classical elementary waves and observe vacuum state for certain initial data. In the limiting process we establish the formation of extreme concentration for a state variable in terms of Dirac delta distribution. Further it is observed that the delta shock solution of the governing system is different from that of pressureless gas dynamics system so a perturbation to the flux is made and the intrinsic phenomena of concentration and cavitation is examined in the limiting case. Additionally we perform numerical simulations to note the effect of van der Waals parameter on the solution of the Riemann problem and to observe the formation of delta shock and vacuum state in the limiting cases.

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.

On Model Two-Dimensional Pressureless Gas Flows: Variational Description and Numerical Algorithm Based on Adhesion Dynamics

Weak solutions of the system of pressureless gas dynamics equations in two dimensions are studied. Theoretical issues are considered, namely, the general mathematical theory of conservation laws for the system is addressed. Emphasis is placed on an important distinctive feature of this system: the emergence of strong density singularities along manifolds of different dimensions. This property is characterized as the formation of a hierarchy of singularities. In earlier application-oriented works (e.g., by A.N. Krayko, et al., including more complicated cases of 3D two-phase flows), this property was studied at the physical level of rigor. In this paper, the formation of a hierarchy of singularities is examined mathematically, since, for example, the existence of a solution with a strong singularity at a point (in the 2D case) is rather difficult to prove rigorously. Accordingly, a special numerical algorithm is used to develop mathematical hypotheses concerning solution behavior. Approaches to the construction of a variational principle for weak solutions are considered theoretically. A numerical algorithm based on approximate adhesion dynamics in the multidimensional case is implemented. The algorithm is tested on several examples (2D Riemann problem) in terms of internal convergence and is compared with mathematical results, including those obtained by other authors.

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.

Riemann problems for the 2-D pressureless gas dynamics with external forces

Abstract This paper is concerned with the pressureless gas dynamics with a space-dependent gravity or a time-dependent friction in two dimensions. The basic Riemann problems with two pieces of constant initial data are considered. By the characteristic analysis, the Riemann problems are constructively solved by two kinds of solutions: vacuum solution and delta-shock solution, which describe the phenomena of cavitation and concentration, respectively. Especially, the generalized Rankine–Hugoniot relations for delta-shock waves are clarified and applied to the Riemann problems.

Initial-boundary value problem for 1D pressureless gas dynamics

The paper considers the system of pressureless gas dynamics in one space dimension. The question of solvability of the initial-boundary value problem is addressed. Using the method of generalized potentials and characteristic triangles, extended to the boundary value case, an explicit way of constructing measure-valued solutions is presented. The prescription of boundary data is shown to depend on the behavior of the generalized potentials at the boundary. We show that the constructed solution satisfies an entropy condition and it conserves mass, whereby mass may accumulate at the boundary. Conservation of momentum again depends on the behavior of the generalized boundary potentials. There is a large amount of literature where the initial value problem for the pressureless gas dynamics model has been studied. To our knowledge, this paper is the first one which considers the initial-boundary value problem.

Concentration and Cavitation in the Vanishing Pressure Limit of Solutions to a 3 × 3 Generalized Chaplygin Gas Equations

The phenomena of concentration and cavitation are identified and analyzed by studying the vanishing pressure limit of solutions to the 3x3 isentropic compressible Euler equations for generalized Chaplygin gas (GCG) with a small parameter. It is rigorously proved that, any Riemann solution containing two shocks and possibly one-contact-discontinuity of the GCG equations converges to a delta-shock solution of the same system as the parameter decreases to a certain critical value. Moreover, as the parameter goes to zero, that is, the pressure vanishes, the limiting solution is just the delta-shock solution of the pressureless gas dynamics (PGD) model, and the intermediate density between the two shocks tends to a weighted δ-measure that forms the delta shock wave; any Riemann solution containing two rarefaction waves and possibly one 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 numerical results are presented to exhibit the processes of concentration and cavitation as the pressure decreases.

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

The exact Riemann solutions for the extended macroscopic production model with an adiabatic exponent are constructed in perfectly explicit forms. The asymptotic limit of Riemann solution consisting of 1-shock wave and 2-contact discontinuity tends to a delta shock solution for the pressureless gas dynamics model under the special over-compressive entropy condition as the adiabatic exponent drops to one. In contrast, the asymptotic limit of Riemann solution composed of 1-rarefaction wave and 2-contact discontinuity tends to the vacuum solution surrounded by two contact discontinuities by letting the adiabatic exponent tend to one, in which the state in the interior of the 1-rarefaction wave fan is developed into vacuum. The intrinsic phenomena of concentration and cavitation are identified and investigated carefully during this limiting process, which displays more complicated and completely different behavior compared with previous literature. In addition, some representative numerical calculations are also provided, which are in well agreement with our theoretical results.

Godunov-type schemes for the pressureless gas dynamics and related models

We study and develop new first-order Godunov-type schemes for the weakly hyperbolic pressureless gas dynamics equations and augmented Burgers’ equations. Each of these systems carries the information of propagation of waves with the same fluid velocity. The goal is achieved by first obtaining an Engquist-Osher (EO) type scheme for the pressureless system and then by enhancing the upwinding information present in the EO-type scheme to construct more accurate Godunov-type schemes. The resulting schemes present a lesser amount of numerical dissipation than existing Jordan decomposition-based Flux Difference Splitting (FDS) schemes recently proposed in [N. K. GARG, Numer. Algorithms, 83 (2020) 1091–1121] and are compact and robust. These schemes are tested on a number of numerical examples for one- and two-dimensional pressureless equations of gas dynamics and then to augmented Burgers’ equations.

The intrinsic phenomena of concentration and cavitation on the Riemann solutions for the perturbed macroscopic production model

The exact solutions of the Riemann problems for the two different perturbed macroscopic production models are considered and constructed, respectively, for all the possible cases. It is found that the asymptotic limits of solutions to the Riemann problem for the first kind of perturbed macroscopic production model do not converge to those of the pressureless gas dynamics model, because the delta shock wave in the limiting situation has different propagation speed and strength from those for the pressureless gas dynamics model. In order to remedy it, the second kind of perturbed macroscopic production model is proposed, whose asymptotic limits of Riemann solutions are identical with those of the pressureless gas dynamics model.

Shadow Wave Tracking Procedure and Initial Data Problem for Pressureless Gas Model

In this paper the new procedure for a construction of an approximated solution to initial data problem for one-dimensional pressureless gas dynamics system is introduced. The procedure is based on solving the Riemann problems and tracking singular wave interactions. For that system the new problem with initial data containing Dirac delta function is solved whenever two waves interact. Use of the shadow waves as singular solutions to such problems enables us to easily solve the interaction problems. That permits us to make a simple extension of the well known Wave Front Tracking algorithm. A non-standard part of the new algorithm is dealing with delta functions as a part of a solution. In the final part of the paper we show that the approximated solution has a subsequence converging to a signed Radon measure.

Solutions with Substance Decay in Pressureless Gas Dynamics Systems

Solutions with Substance Decay in Pressureless Gas Dynamics Systems

Weak Asymptotics Method Approach to the Problem of $$\delta$$-Shock Wave Interactions

In this note, we present an unusual picture of interaction of singularities for pressureless gas dynamics.

A relaxation model for numerical approximations of the multidimensional pressureless gas dynamics system

Relaxation models for the pressureless gas dynamics (PGD) equations attempt to satisfy the strictly hyperbolic conservation law in order to employ the well-posed approximated Riemann solvers. In this study, a new type of relaxation model is proposed to resolve two shortcomings of the existing relaxation models: the constant propagation speed of sound, and the collapse of delta shock waves in multidimensional problems. The proposed model seeks a strictly hyperbolic system of equations without any special consideration for the proper values of the propagation speed of sound. Numerical tests showed that the proposed model can accurately describe the behavior of the PGD equations, in particular, the occurrence of delta shock waves and vacuum states in a multidimensional problem.

Solutions with Substance Decay in Pressureless Gas Dynamics Systems

Solutions with Substance Decay in Pressureless Gas Dynamics Systems

An implementation of the weighted essential non-oscillatory scheme for numerical approximations of the pressureless gas dynamics equations

In this study, the Weighted Essential Non-Oscillatory (WENO) scheme is implemented for numerical approximations of the pressureless gas dynamics equations. The Harten-Lax-van Leer-Contact (HLLC) approximate Riemann solver serves as a basic building block with the fifth order WENO scheme involving the first order Lax-Friedrichs scheme as a flux limiter for the numerical stability. In particular, the WENO scheme suppressed the oscillatory around the delta shock waves and the flux limiter guaranteed the positivity of densities around the vacuum. Lastly, numerical testes for one-dimensional problems are presented for capturing the delta shock waves and vacuum.

Interaction of δ-Shock Waves in a System of Pressureless Gas Dynamics Equations

In the present paper, an analytic description of nonlinear delta-shock waves interaction is described by using the weak asymptotics method.

Detailed description of the evolution mechanism for singularities in the system of pressureless gas dynamics

The system of pressureless gas dynamics is a hydrodynamically justified generalization of the system consisting of the Burgers vector equation in the limit of vanishing viscosity and the mass conservation law. The latter system of equations was intensively used, in particular, in astrophysics to describe the large scale structure of the Universe. The solutions of the vector Burgers equation involve interesting dynamics of singularities, which can describe concentration processes. However, this dynamics does not satisfy the law of momentum conservation, which prevents us from treating it as dynamics of material objects. In this paper, momentum-conserving dynamics of singularities is investigated on the basis of the pressureless gas dynamics system. Such dynamics turns out to be more diverse and complex, but it is also possible to formulate a variational approach, for which the basic principles and relations are obtained in the work.