scholarly journals The Perturbed Riemann Problem with Delta Shock for a Hyperbolic System

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Xinli Han ◽  
Lijun Pan
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Hyperbolic System ◽  
Contact Discontinuity ◽  
Riemann Problems ◽  
Delta Shock ◽  
Riemann Solution ◽  
Internal Mechanism

In this paper, we study the perturbed Riemann problem with delta shock for a hyperbolic system. The problem is different from the previous perturbed Riemann problems which have no delta shock. The solutions to the problem are obtained constructively. From the solutions, we see that a delta shock in the corresponding Riemann solution may turn into a shock and a contact discontinuity under a perturbation of the Riemann initial data. This shows the instability and the internal mechanism of a delta shock. Furthermore, we find that the Riemann solution of the hyperbolic system is instable under this perturbation, which is also quite different from the previous perturbed Riemann problems.

Measure-Valued Solutions to a Non-Strictly Hyperbolic System with Delta-Type Riemann Initial Data

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuangrong Li ◽  
Chun Shen
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Riemann Problem ◽  
Contact Discontinuity ◽  
Initial Condition ◽  
Global Measure ◽  
Delta Shock Wave ◽  
Dirac Delta ◽  
Delta Shock ◽  
Delta Type

AbstractThis paper is concerned with the construction of global measure-valued solutions to the extended Riemann problem for a non-strictly hyperbolic system of two conservation laws with delta-type initial data. The wave interaction problems have been extensively studied for all kinds of situations by using the initial condition consisting of constant states in three pieces instead of delta-type initial data under the perturbation method. The measure-valued solutions of the extended Riemann problem are achieved constructively when the perturbed parameter tends to zero. During the process of constructing solutions, a new and interesting nonlinear phenomenon is discovered, in which the initial Dirac delta function travels along the trajectory of either delta shock wave or contact discontinuity (or delta contact discontinuity). Moreover, a delta shock wave is separated into a delta contact discontinuity and a shock wave during the process of delta shock wave penetrating a composite wave composed of a rarefaction wave and a contact discontinuity. In addition, we further consider the constructions of global measure-valued solutions when the initial condition contains Dirac delta functions at two different initial points.

scholarly journals The Vanishing Pressure Limit of Riemann Solutions to the Non-Isentropic Euler Equations for Generalized Chaplygin Gas

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qixia Ding ◽  
Lihui Guo
Keyword(s):  
Shock Wave ◽  
Contact Discontinuity ◽  
Vacuum State ◽  
Chaplygin Gas ◽  
Delta Shock Wave ◽  
Generalized Chaplygin Gas ◽  
Riemann Solutions ◽  
Pressure Limit ◽  
Delta Shock ◽  
Riemann Solution

We analyze the appearance of delta shock wave and vacuum state in the vanishing pressure limit of Riemann solutions to the non-isentropic generalized Chaplygin gas equations. As the pressure vanishes, the Riemann solution including two shock waves and possible one contact discontinuity converges to a delta shock wave solution. Both the densityρand the internal energyHsimultaneously present a Dirac delta singularity. And the Riemann solution involving two rarefaction waves and possible one contact discontinuity converges to a solution involving vacuum state of the transport equations.

scholarly journals Constructive Approach of the Solution of Riemann Problem for Shallow Water Equations with Topography and Vegetation

2020 ◽  
Vol 28 (2) ◽  
pp. 93-114
Author(s):  
Stelian Ion ◽  
Stefan-Gicu Cruceanu ◽  
Dorin Marinescu
Keyword(s):  
Initial Data ◽  
Shallow Water ◽  
Global Solution ◽  
Riemann Problem ◽  
Hyperbolic System ◽  
Shallow Water Equations ◽  
Local Existence ◽  
Water Model ◽  
Constructive Approach ◽  
Shallow Water Model

AbstractWe investigate the Riemann Problem for a shallow water model with porosity and terrain data. Based on recent results on the local existence, we build the solution in the large settings (the magnitude of the jump in the initial data is not supposed to be “small enough”). One di culty for the extended solution arises from the double degeneracy of the hyperbolic system describing the model. Another di culty is given by the fact that the construction of the solution assumes solving an equation which has no global solution. Finally, we present some cases to illustrate the existence and non-existence of the solution.

scholarly journals Four-Quadrant Riemann Problem for a 2×2 System II

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 592
Author(s):  
Jinah Hwang ◽  
Suyeon Shin ◽  
Myoungin Shin ◽  
Woonjae Hwang
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Wave Interactions ◽  
Delta Shock ◽  
Elementary Wave ◽  
Initial Discontinuity ◽  
The Rich ◽  
Four Quadrant

In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.

scholarly journals Riemann Problem for Shallow Water Equation with Vegetation

2018 ◽  
Vol 26 (2) ◽  
pp. 145-173
Author(s):  
Stelian Ion ◽  
Dorin Marinescu ◽  
Stefan-Gicu Cruceanu
Keyword(s):  
Shallow Water ◽  
Riemann Problem ◽  
Shallow Water Equation ◽  
Water Type ◽  
Riemann Problems ◽  
Riemann Solution ◽  
Surface System ◽  
Family Based

Abstract We investigate the existence of the solution of the Riemann Problem for a simplified water ow model on a vegetated surface - system of shallow water type equations. It is known that the system with discontinuous topography is non-conservative even if the porosity is absent. A system with continuous topography and discontinuous porosity is also non-conservative. In order to define Riemann solution for such systems, it is necessary to introduce a family of paths that connects the states defining the Riemann Problem. We focus our attention towards choosing such a family based on physical arguments. We provide the structure of the solution for such Riemann Problems.

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

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.

scholarly journals Solutions with concentration and cavitation to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas

2019 ◽  
Vol 17 (1) ◽  
pp. 220-241 ◽  
Author(s):  
Yunfeng Zhang ◽  
Meina Sun ◽  
Xiuli Lin
Keyword(s):  
Riemann Problem ◽  
Vacuum State ◽  
Chaplygin Gas ◽  
Euler System ◽  
Phase Plane Analysis ◽  
Delta Shock Wave ◽  
Plane Analysis ◽  
Delta Shock ◽  
Riemann Solution ◽  
Shock Wave Solution

Abstract The solutions to the Riemann problem for the isentropic relativistic Euler system for the extended Chaplygin gas are constructed for all kinds of situations by using the method of phase plane analysis. The asymptotic limits of solutions to the Riemann problem for the relativistic extended Chaplygin Euler system are investigated in detail when the pressure given by the equation of state of extended Chaplygin gas becomes that of the pressureless gas. During the process of vanishing pressure, the phenomenon of concentration can be identified and analyzed when the two-shock Riemann solution tends to a delta shock wave solution as well as the phenomenon of cavitation also being captured and observed when the two-rarefaction-wave Riemann solution tends to a two-contact-discontinuity solution with a vacuum state between them.

scholarly journals A Deposition Model: Riemann Problem and Flux-Function Limits of Solutions

2018 ◽  
Vol 2018 ◽  
pp. 1-14
Author(s):  
Hongjun Cheng ◽  
Shiwei Li
Keyword(s):  
Initial Data ◽  
Riemann Solutions ◽  
Limit System ◽  
Flux Function ◽  
Deposition Model ◽  
Delta Shock ◽  
Riemann Solution ◽  
Scalar Conservation Law ◽  
Vacuum States ◽  
Scalar Conservation

The Riemann solutions of a deposition model are shown. A singular flux-function limit of the obtained Riemann solutions is considered. As a result, it is shown that the Riemann solutions of the deposition model just converge to the Riemann solutions of the limit system, the scalar conservation law with a linear flux function involving discontinuous coefficient. Especially, for some initial data, the two-shock Riemann solution of the deposition model tends to the delta-shock Riemann solution of the limit system; by contrast, for some initial data, the two-rarefaction-wave Riemann solution of the deposition model tends to the vacuum Riemann solution of the limit system. Some numerical results exhibiting the formation processes of delta-shocks and vacuum states are presented.

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

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5355-5373 ◽  
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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