Construction of the Global Solutions to the Perturbed Riemann Problem for the Leroux System

The global solutions of the perturbed Riemann problem for the Leroux system are constructed explicitly under the suitable assumptions when the initial data are taken to be three piecewise constant states. The wave interaction problems are widely investigated during the process of constructing global solutions with the help of the geometrical structures of the shock and rarefaction curves in the phase plane. In addition, it is shown that the Riemann solutions are stable with respect to the specific small perturbations of the Riemann initial data.

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The Perturbed Riemann Problem for Special Keyfitz-Kranzer System with Three Piecewise Constant States

Advances in Mathematical Physics ◽

10.1155/2017/3276182 ◽

2017 ◽

Vol 2017 ◽

pp. 1-12

Author(s):

Yuhao Jiang ◽

Chun Shen

Keyword(s):

Initial Data ◽

Riemann Problem ◽

Global Solutions ◽

Wave Interactions ◽

Small Perturbations ◽

Riemann Solutions ◽

Piecewise Constant

The Riemann problem for a special Keyfitz-Kranzer system is investigated and then seven different Riemann solutions are constructed. When the initial data are chosen as three piecewise constant states under suitable assumptions, the global solutions to the perturbed Riemann problem are constructed explicitly by studying all occurring wave interactions in detail. Furthermore, the stabilities of solutions are obtained under the specific small perturbations of Riemann initial data.

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Riemann problem with delta initial data for the two-dimensional steady pressureless isentropic relativistic Euler equations

Mathematical Modelling of Natural Phenomena ◽

10.1051/mmnp/2021011 ◽

2021 ◽

Author(s):

Yu Zhang ◽

Yanyan Zhang

Keyword(s):

Initial Data ◽

Euler Equations ◽

Riemann Problem ◽

Contact Discontinuity ◽

Global Solutions ◽

Two Dimensional ◽

Relativistic Euler Equations ◽

Riemann Solutions ◽

Isentropic Relativistic Euler Equations ◽

The Stability

The Riemann problem for the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data is studied. First, the perturbed Riemann problem with three pieces constant initial data is solved. Then, via discussing the limits of solutions to the perturbed Riemann problem, the global solutions of Riemann problem with delta initial data are completely constructed under the stability theory of weak solutions. Interestingly, the delta contact discontinuity is found in the Riemann solutions of the two-dimensional steady pressureless isentropic relativistic Euler equations with delta initial data.

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Analysis of the Stability of the Riemann Problem for a Simplified Model in Magnetogasdynamics

Advances in Mathematical Physics ◽

10.1155/2016/7526413 ◽

2016 ◽

Vol 2016 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Yujin Liu ◽

Wenhua Sun

Keyword(s):

Shock Wave ◽

Riemann Problem ◽

Contact Discontinuity ◽

Ideal Gas ◽

Simplified Model ◽

Characteristic Method ◽

Small Perturbations ◽

Riemann Solutions ◽

One Dimensional ◽

The Stability

The generalized Riemann problem for a simplified model of one-dimensional ideal gas in magnetogasdynamics in a neighborhood of the origin(t>0)in the(x,t)plane is considered. According to the different cases of the corresponding Riemann solutions, we construct the perturbed solutions uniquely with the characteristic method. We find that, for some case, the contact discontinuity appears after perturbation while there is no contact discontinuity of the corresponding Riemann solution. For most cases, the Riemann solutions are stable and the perturbation can not affect the corresponding Riemann solutions. While, for some few cases, the forward (backward) rarefaction wave can be transformed into the forward (backward) shock wave which shows that the Riemann solutions are unstable under such local small perturbations of the Riemann initial data.

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The Existence of Zero Waves for Nonsimplified Chromatography System

Mathematical Problems in Engineering ◽

10.1155/2020/1970505 ◽

2020 ◽

Vol 2020 ◽

pp. 1-16

Author(s):

Maozhou Lin ◽

Lihui Guo ◽

Yinsong Bai

Keyword(s):

Shock Wave ◽

Initial Data ◽

Phase Plane ◽

Wave Interaction ◽

Phase Plane Analysis ◽

Delta Shock Wave ◽

Plane Analysis ◽

Delta Shock ◽

The Right ◽

Good Agreement

In this paper, we mainly consider Riemann problem for the widely used nonsimplified chromatography system with initial data consisting of three pieces of constant states. Through phase plane analysis, the solutions of the nonsimplified chromatography system are established. When the different initial data tend to −1 from the right side, the existence of zero shock wave, zero delta shock wave, and zero rarefaction wave is obtained via analyzing its wave interaction. Finally, the correctness of the main conclusions is verified by numerical simulation, and the numerical results are in good agreement with the theoretical solutions of several experimental cases.

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EXACT RIEMANN SOLUTIONS TO COMPRESSIBLE EULER EQUATIONS IN DUCTS WITH DISCONTINUOUS CROSS-SECTION

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891612500130 ◽

2012 ◽

Vol 09 (03) ◽

pp. 403-449 ◽

Cited By ~ 16

Author(s):

EE HAN ◽

MAREN HANTKE ◽

GERALD WARNECKE

Keyword(s):

Initial Data ◽

Euler Equations ◽

Cross Section ◽

Phase Plane ◽

Compressible Euler Equations ◽

Crucial Point ◽

Riemann Solutions ◽

Compressible Euler ◽

Velocity Pressure ◽

Pressure Phase

We determine completely the exact Riemann solutions for the system of Euler equations in a duct with discontinuous varying cross-section. The crucial point in solving the Riemann problem for hyperbolic system is the construction of the wave curves. To address the difficulty in the construction due to the nonstrict hyperbolicity of the underlying system, we introduce the L-M and R-M curves in the velocity-pressure phase plane. The behaviors of the L-M and R-M curves for six basic cases are fully analyzed. Furthermore, we observe that in certain cases the L-M and R-M curves contain the bifurcation which leads to the nonuniqueness of the Riemann solutions. Nevertheless, all possible Riemann solutions including classical as well as resonant solutions are solved in a uniform framework for any given initial data.

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The Perturbed Riemann Problem for the Aw-Rascle Model with Modified Chaplygin Gas Pressure

Advances in Mathematical Physics ◽

10.1155/2018/7104527 ◽

2018 ◽

Vol 2018 ◽

pp. 1-7

Author(s):

Yujin Liu ◽

Wenhua Sun

Keyword(s):

Riemann Problem ◽

Large Time ◽

Chaplygin Gas ◽

Global Structure ◽

Gas Pressure ◽

Modified Chaplygin Gas ◽

Riemann Solutions ◽

Asymptotic Behaviors ◽

Piecewise Constant ◽

The Stability

This paper is concerned with the perturbed Riemann problem for the Aw-Rascle model with the modified Chaplygin gas pressure. We obtain constructively the solutions when the initial values are three piecewise constant states. The global structure and the large-time asymptotic behaviors of the solutions are discussed case by case. Further, we obtain the stability of the corresponding Riemann solutions as the initial perturbed parameter tends to zero.

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Interactions of Delta Shock Waves for Zero-Pressure Gas Dynamics with Energy Conservation Law

Advances in Mathematical Physics ◽

10.1155/2016/1783689 ◽

2016 ◽

Vol 2016 ◽

pp. 1-12

Author(s):

Wei Cai ◽

Yanyan Zhang

Keyword(s):

Theoretical Analysis ◽

Shock Waves ◽

Initial Data ◽

Gas Dynamics ◽

Riemann Solutions ◽

Piecewise Constant ◽

Delta Shock ◽

Vacuum States ◽

Delta Shock Waves ◽

Energy Conservation Law

We study the interactions of delta shock waves and vacuum states for the system of conservation laws of mass, momentum, and energy in zero-pressure gas dynamics. The Riemann problems with initial data of three piecewise constant states are solved case by case, and four different configurations of Riemann solutions are constructed. Furthermore, the numerical simulations completely coinciding with theoretical analysis are shown.

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On the delta shockwave interactions for the isentropic Chaplygin gas system consisting of three scalar equations

Filomat ◽

10.2298/fil1916355s ◽

2019 ◽

Vol 33 (16) ◽

pp. 5355-5373 ◽

Cited By ~ 2

Author(s):

Meina Sun ◽

Jie Xin

Keyword(s):

Shock Wave ◽

Initial Data ◽

Riemann Problem ◽

Propagation Speed ◽

Chaplygin Gas ◽

Euler System ◽

Delta Shock Wave ◽

Riemann Solutions ◽

Delta Shock ◽

Scalar Equations

The Riemann problem for the one-dimensional version of isentropic compressible Euler system for the Chaplygin gas consisting of three scalar equations is considered. It is shown that the Riemann solutions involve only two situations: the combination of three contact discontinuities or a delta shock wave. The generalized Rankine-Hugoniot conditions of delta shock wave are derived and the exact delta shock wave solution including the strength and propagation speed is obtained explicitly. The solutions to the perturbed Riemann problem are constructed globally when the initial data are taken to be the three piecewise constant initial data. The wave interaction problem is extensively investigated and some interesting phenomena are observed. It is shown that the limits of solutions to the perturbed Riemann problem converge to the corresponding ones to the Riemann problem when the perturbation parameter tends to zero.

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