scholarly journals Nonlinear algebraic analysis of delta shock wave solutions to Burgers’ equation

2003 ◽  
Vol 210 (1) ◽  
pp. 165-187 ◽  
Author(s):  
J.-A. Marti
Keyword(s):  
Shock Wave ◽  
Burgers Equation ◽  
Delta Shock Wave ◽  
Algebraic Analysis ◽  
Delta Shock ◽  
Wave Solutions

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.

Riemann problem and limits of solutions to the isentropic relativistic Euler equations for isothermal gas with flux approximation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yu Zhang ◽  
Yanyan Zhang
Keyword(s):  
Shock Wave ◽  
Euler Equations ◽  
Riemann Problem ◽  
Delta Shock Wave ◽  
Relativistic Euler Equations ◽  
Delta Shock ◽  
Isentropic Relativistic Euler Equations ◽  
Flux Approximation ◽  
Two Parameter ◽  
Pressureless Relativistic Euler Equations

Abstract We are concerned with the vanishing flux-approximation limits of solutions to the isentropic relativistic Euler equations governing isothermal perfect fluid flows. The Riemann problem with a two-parameter flux approximation including pressure term is first solved. Then, we study the limits of solutions when the pressure and two-parameter flux approximation vanish, respectively. It is shown that, any two-shock-wave Riemann solution converges to a delta-shock solution of the pressureless relativistic Euler equations, and the intermediate density between these two shocks tends to a weighted δ-measure that forms a delta shock wave. By contract, any two-rarefaction-wave solution tends to a two-contact-discontinuity solution of the pressureless relativistic Euler equations, and the intermediate state in between tends to a vacuum state.

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