A distributional product approach to the delta shock wave solution for the one-dimensional zero-pressure gas dynamics system

2018 ◽  
Vol 105 ◽  
pp. 105-112 ◽  
Author(s):  
Chun Shen ◽  
Meina Sun
Keyword(s):  
Shock Wave ◽  
Wave Solution ◽  
Gas Dynamics ◽  
Delta Shock Wave ◽  
One Dimensional ◽  
Delta Shock ◽  
Shock Wave Solution ◽  
Distributional Product ◽  
Product Approach ◽  
The One

scholarly journals Limits of Riemann Solutions to the Relativistic Euler Systems for Chaplygin Gas as Pressure Vanishes

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Gan Yin ◽  
Kyungwoo Song
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Wave Solution ◽  
Chaplygin Gas ◽  
Euler System ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Euler Systems ◽  
Delta Shock ◽  
Shock Wave Solution

Vanishing pressure limits of Riemann solutions to relativistic Euler system for Chaplygin gas are identified and analyzed in detail. Unlike the polytropic or barotropic gas case, as the parameter decreases to a critical value, the two-shock solution converges firstly to a delta shock wave solution to the same system. It is shown that, as the parameter decreases, the strength of the delta shock increases. Then as the pressure vanishes ultimately, the solution is nothing but the delta shock wave solution to the zero pressure relativistic Euler system. Meanwhile, the two-rarefaction wave solution and the solution containing one-rarefaction wave and one-shock wave tend to the vacuum solution and the contact discontinuity solution to the zero pressure relativistic Euler system, respectively.

A DISTRIBUTIONAL PRODUCT APPROACH TO δ-SHOCK WAVE SOLUTIONS FOR A GENERALIZED PRESSURELESS GAS DYNAMICS SYSTEM

2014 ◽  
Vol 25 (01) ◽  
pp. 1450007 ◽  
Author(s):  
C. O. R. SARRICO
Keyword(s):  
Shock Wave ◽  
Gas Dynamics ◽  
General Framework ◽  
Solution Concept ◽  
Singular Solutions ◽  
Wave Solutions ◽  
Shock Wave Solution ◽  
Consistent Extension ◽  
Distributional Product ◽  
Product Approach

In the setting of a product of distributions which is not defined by approximation processes, we are able to consider a Riemann problem for the system ut + [ϕ(u)]x = 0, vt + [ψ(u)v]x = 0, with the unknown states u, v in convenient spaces of distributions and ϕ, ψ : ℝ → ℝ continuous. A consistent extension of the classical solution concept will show the possible arising of a δ-shock wave solution. This procedure affords a simpler and more general framework to construct singular solutions and can surely be applied to other equations or systems.

scholarly journals Concentration in vanishing adiabatic exponent limit of solutions to the Aw–Rascle traffic model

2021 ◽  
pp. 1-35
Author(s):  
Shouqiong Sheng ◽  
Zhiqiang Shao
Keyword(s):  
Shock Wave ◽  
Theoretical Analysis ◽  
Shock Waves ◽  
Gas Dynamics ◽  
Traffic Model ◽  
Adiabatic Exponent ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Distribution Sense ◽  
Delta Shock

In this paper, we study the phenomenon of concentration and the formation of delta shock wave in vanishing adiabatic exponent limit of Riemann solutions to the Aw–Rascle traffic model. It is proved that as the adiabatic exponent vanishes, the limit of solutions tends to a special delta-shock rather than the classical one to the zero pressure gas dynamics. In order to further study this problem, we consider a perturbed Aw–Rascle model and proceed to investigate the limits of solutions. We rigorously proved that, as the adiabatic exponent tends to one, any Riemann solution containing two shock waves tends to a delta-shock to the zero pressure gas dynamics in the distribution sense. Moreover, some representative numerical simulations are exhibited to confirm the theoretical analysis.

scholarly journals Delta Shock Wave for the Suliciu Relaxation System

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Richard De la cruz ◽  
Juan Galvis ◽  
Juan Carlos Juajibioy ◽  
Leonardo Rendón
Keyword(s):  
Riemann Problem ◽  
Entropy Condition ◽  
Delta Shock Wave ◽  
One Dimensional ◽  
Hugoniot Relation ◽  
Delta Shock ◽  
The Cauchy Problem ◽  
The One ◽  
Linearly Degenerate ◽  
Characteristic Field

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered3×3system is such that every characteristic field is linearly degenerate. We show an explicit solution for the Cauchy problem with initial data inL∞. We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.

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

2021 ◽  
Author(s):  
Yunfeng Zhang ◽  
Meina Sun
Keyword(s):  
Shock Wave ◽  
Exact Solutions ◽  
Gas Dynamics ◽  
Propagation Speed ◽  
Production Model ◽  
Dynamics Model ◽  
Delta Shock Wave ◽  
Riemann Solutions ◽  
Delta Shock ◽  
Production Models

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 coverage 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.

scholarly journals On the evolution of shock-waves in mathematical models of the aorta

1981 ◽  
Vol 22 (3) ◽  
pp. 257-269 ◽  
Author(s):  
L. K. Forbes
Keyword(s):  
Shock Wave ◽  
Blood Flow ◽  
Shock Waves ◽  
Gas Dynamics ◽  
Pulse Propagation ◽  
One Dimensional ◽  
Flow Models ◽  
Wave Development ◽  
Blood Flow Models ◽  
The One

AbstractThe one-dimensional, non-linear theory of pulse propagation in large arteries is examined in the light of the analogy which exists with gas dynamics. Numerical evidence for the existence of shock-waves in current one-dimensional blood-flow models is presented. Some methods of suppressing shock-wave development in these models are indicated.

Sign in / Sign up

Export Citation Format

Share Document