Interactions Of Elementary Waves Research Articles

Interactions between elementary waves and weak discontinuity in two-layer blood flow through artery

Blood flow through arterial circulation can be characterized by fluid transport in flexible tubes and modeled mathematically using the conservation of mass and momentum. A one-dimensional model for two-layer blood flow with different blood velocities and the same constant density in each layer derived from the Euler equations of gas dynamics by taking the vertical average across each layer. This work presents interactions of elementary waves with a weak discontinuity for the quasilinear 3 × 3 system of conservation laws governing the two-layer blood flow in arteries. Exploiting elementary waves as a single-parameter curve, we study the Riemann solution uniquely and consequently establish the condition on initial data for the existence of a solution to the Riemann problem. Furthermore, we discuss the evolution of weak discontinuity waves and subsequently derive their amplitudes; in what follows, we investigate the interactions of weak discontinuity with contact discontinuity and shocks. Finally, a series of numerical tests have been performed to understand the impact of shock strength and the initial data on the amplitudes of reflected and transmitted waves and the jumps in shock acceleration.

On the Riemann problem and interaction of elementary waves for two‐layered blood flow model through arteries

In this paper, we focus on the Riemann problem for two‐layered blood flow model, which is represented by a system of quasi‐linear hyperbolic partial differential equations (PDEs) derived from the Euler equations by vertical averaging across each layer. We consider the Riemann problem with varying velocities and equal constant density through arteries. For instance, the flow layer close to the wall of vessel has a slower average speed than the layer far from the vessel because of the viscous effect of the blood vessel. We first establish the existence and uniqueness of the corresponding Riemann solution by a thorough investigation of the properties of elementary waves, namely, shock wave, rarefaction wave, and contact discontinuity wave. Further, we extensively analyze the elementary wave interaction between rarefaction wave and shock wave with contact discontinuity and rarefaction wave and shock wave. The global structure of the Riemann solutions after each wave interaction is explicitly constructed and graphically illustrated towards the end.

The Riemann problem and a Godunov-type scheme for a traffic flow model on two lanes with two velocities

We establish and investigate a traffic flow model on two lanes with two velocities in this paper. The model takes account of the conservation of mass and acceleration equation on each lane. We derive the elementary waves and solve the Riemann problem of the new model in 1-D case constructively. The interaction of elementary waves are then discussed subsequently. Besides, we develop a Godunov-type scheme with an exact Riemann solver established on the Riemann solutions. Numerical simulations generated from the real traffic flow situations are given to verify the effectiveness and versatility of the model.

Interaction of elementary waves for the Aw–Rascle traffic flow model with variable lane width

Interaction of elementary waves for the Aw–Rascle traffic flow model with variable lane width

Nonlinear Wave Interactions in a Macroscopic Production Model

In this article, we study the exhaustive analysis of nonlinear wave interactions for a 2 × 2 homogeneous system of quasilinear hyperbolic partial differential equations (PDEs) governing the macroscopic production. We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations. Furthermore, we study the interaction between simple waves in detail through exact solution of general initial value problem. Finally, we discuss the all possible interaction of elementary waves using the solution of Riemann problem.

Wave Interactions for Chaplygin Gas with Combustion

The elementary wave interactions for the Chapman-Jouguet model with combustion are investigated. We obtain the unique solution of the initial value problem under the global entropy conditions. We analyze the elementary wave interactions in the phase plane and construct uniquely the solution of this initial value problem. It is found that the combustion wave solution of the corresponding Riemann may be extinguished after perturbation which shows that the unburnt gas is unstable.

Wave Interactions of the Aw-Rascle Model for Generalized Chaplygin Gas

In this paper, we investigate the elementary wave interactions of the Aw-Rascle model for the generalized Chaplygin gas. We construct the unique solution by the characteristic analysis method and obtain the stability of the corresponding Riemann solutions under such small perturbations on the initial values. We find that the elementary wave interactions have a much more simple structure for Temple class than general systems of conservation laws. It is important to study the elementary waves interactions of the traffic flow system for the generalized Chaplygin gas not only because of their significance in practical applications in the traffic flow system, but also because of their basic role for the general mathematical theory.

Wave Interactions for the Suliciu Relaxation System

In this paper, we investigate the elementary wave interactions for the Suliciu relaxation system and construct uniquely the solution by the characteristic analysis method in the phase plane. We find that the elementary wave interactions have a much simpler structure for the Temple class than the general systems of conservation laws. It is observed that the Riemann solutions of the Suliciu relaxation system are stable under the small perturbation on the Riemann initial data.

The interaction of elementary waves for nonisentropic flow in a variable cross-section duct

The interaction of elementary waves for isentropic flow in a variable cross-section duct is obtained (\cite{ShengZhang}). The authors have discussed rarefaction wave or shock wave interacts with stationary wave. In this paper, we extend their results to the nonisentropic flow. It can be proved that if one changes $(u,\rho)$ plane in isentropic flow with $(u,p)$ plane in nonisentropic case, the interaction results of rarefaction wave or shock wave with stationary wave can be moved parallel from the isentropic case when contact discontinuity is involved. Thus we mainly focus on the interactions between contact discontinuity and stationary wave. Some numerical results are given to verify our analysis. The results can apply to the interaction of more complicated wave patterns.

The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas

The exact Riemann solutions for a macroscopic production model under the equation of state given by the Chaplygin gas are solved explicitly for all possible Riemann initial data. It is discovered interestingly that a composite hyperbolic wave is involved in Riemann solution under some specially designated initial conditions, which is made up of a rarefaction wave and a delta contact discontinuity attached on the wave front of the rarefaction wave. Furthermore, the constructions of global solutions to the perturbed Riemann problem for this system are also displayed in completely explicit forms when the initial data are taken to be three piecewise constant states under some suitable restrictive conditions by using the method of characteristics. During the process of constructing global solutions, the interactions of elementary waves are studied in detail. Moreover, it is proved rigorously that Riemann solutions are stable with respect to the specific small perturbations of Riemann initial data.

Riemann problem and wave interactions for the one-dimensional relativistic string equation in Minkowski space

In this paper, we consider the Riemann problem for the one-dimensional relativistic string equation in Minkowski space. It is shown that the delta shock wave will be constructed in the Riemann solutions in some certain situations. Furthermore, the interactions of elementary waves including contact discontinuity and delta shock wave are obtained.

Interaction of elementary waves with a weak discontinuity in an isothermal drift-flux model of compressible two-phase flows

In this paper, we study the interaction of elementary waves of the Riemann problem with a weak discontinuity for an isothermal no-slip compressible gas-liquid drift flux equation of two-phase flows. We construct the solution of the Riemann problem in terms of a one parameter family of curves. Using the properties of elementary waves, we prove a necessary and sufficient condition on initial data for which the solution of the Riemann problem consists of a left shock, contact discontinuity, and a right shock. Moreover, we derive the amplitudes of weak discontinuity and discuss the interactions of weak discontinuity with shocks and contact discontinuity. Finally, we carry out some tests to investigate the effect of shock strength and initial data on the jump in shock acceleration and the amplitudes of reflected and transmitted waves.

Elementary Wave Interactions for the Simplified Combustion Model in Magnetogasdynamics

We investigate the elementary wave interactions for the simplified combustion model in magnetogasdynamics. Under the modified entropy conditions, we construct the unique solution and observe some interesting phenomena; such as, the combustion wave may be extinguished by the contact discontinuity or the shock wave. Especially, the transition between the detonation wave and the deflagration wave is also captured.

Riemann problem and elementary wave interactions in dusty gas

The present paper concerns with the study of the Riemann problem for a quasi-linear hyperbolic system of partial differential equations governing the one dimensional isentropic dusty gas flow. The shock and rarefaction waves and their properties for the problem are investigated. We also examine how some of the properties of shock and rarefaction waves in a dusty gas flow differ from isentropic ideal gas flow. The solution of Riemann problem of dusty gas flow for different initial data is discussed. Under certain conditions, the uniqueness and existence of the solution of the Riemann problem has been analyzed. Finally, all possible interactions of elementary waves are discussed.

Global Solution of a Nonlinear Conservation Law with Weak Discontinuous Flux in the Half Space

This paper is concerned with the initial-boundary value problem of a nonlinear conservation law in the half space R+= {x |x > 0} where a>0 , u(x,t) is an unknown function of x ∈ R+ and t>0 , u ± , um are three given constants satisfying um=u+≠u- or um=u-≠u+ , and the flux function f is a given continuous function with a weak discontinuous point ud. The main purpose of our present manuscript is devoted to studying the structure of the global weak entropy solution for the above initial-boundary value problem under the condition of f '-(ud) > f '+(ud). By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial-boundary value problem, and investigate the interaction of elementary waves with the boundary and the boundary behavior of the weak entropy solution.

Elementary wave interactions in blood flow through artery

In this paper, we consider the Riemann problem and interaction of elementary waves for the quasilinear hyperbolic system of conservation laws that arises in blood flow through arteries. We study the properties of solution involving shocks and rarefaction waves and establish the existence and uniqueness conditions. We show that the Riemann problem is solvable for arbitrary initial data under certain condition and construct the condition for no-feasible solution. Finally, we present numerical examples with different initial data and discuss all possible interactions of elementary waves.

Construction of Global Weak Entropy Solution of Initial-Boundary Value Problem for Scalar Conservation Laws with Weak Discontinuous Flux

This paper is concerned with the initial-boundary value problem of scalar conservation laws with weak discontinuous flux, whose initial data are a function with two pieces of constant and whose boundary data are a constant function. Under the condition that the flux function has a finite number of weak discontinuous points, by using the structure of weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux-Nedelec, we give a construction method to the global weak entropy solution for this initial-boundary value problem, and by investigating the interaction of elementary waves and the boundary, we clarify the geometric structure and the behavior of boundary for the weak entropy solution.

Wave Interactions in Non-ideal Isentropic Magnetogasdynamics

In this paper, we consider the Riemann problem and wave interactions for a quasi-linear hyperbolic system of partial differential equations governing the one dimensional unsteady simple wave flow of an isentropic, non-ideal, inviscid and perfectly conducting compressible fluid, subject to a transverse magnetic field. This class of equations includes, as a special case of ideal isentropic magnetogasdynamics. We study the shock and rarefaction waves and their properties, and show the existence and uniqueness of the solution to the Riemann problem for arbitrary initial data under certain conditions and then we discuss the vacuum state in non-ideal isentropic magnetogasdynamics. We discuss numerical tests and study the solution influenced by the van der Waals excluded volume for different initial data along with all possible interactions of elementary waves.

Elementary wave interactions in magnetogasdynamics

This paper is mainly concerned with the interactions of the elementary waves for the one-dimensional ideal Magnetogasdynamics with transverse magnetic field. By applying the method of the characteristic analysis, we obtain constructively the solutions of the all possible wave interactions when the initial data are three piecewise constant states. We find that the result is very different from that of the corresponding case of the conventional gas dynamics. However, the result is consistent with that of the corresponding case for Euler equations when the magnetic field vanishes.

Elementary Wave Interactions to the Compressible Euler Equations for Chaplygin Gas in Two Dimensions

In order to construct global piecewise smooth solutions to two-dimensional (2D) Riemann problems for the compressible Euler equations, it is important to investigate 2D elementary wave interactions. Recently, 2D shock interactions and 2D Riemann problems for the compressible Euler equations for Chaplygin gas have been investigated. In this paper, we study several types of 2D elementary wave interactions of the Chaplygin gas Euler equations. These elementary waves include shock waves, simple waves, and delta waves. Using these elementary wave interactions, global piecewise smooth solutions to several 2D Riemann problems are constructed.