Structural stability of the Riemann solution for a strictly hyperbolic system of conservation laws with flux approximation

Abstract

In this article, we study the Riemann problem for a strictly hyperbolic system of conservation laws under the linear approximation of flux functions with three parameters. The approximation does not affect the structure of Riemann problem. Furthermore, we prove that the Riemann solution to the approximated system converges to the original system as the perturbation parameter tends to zero.

Delta shock wave and wave interactions in a thin film of a perfectly soluble anti-surfactant solution

“…The Riemann problem of (1) cannot be solved for all possible Riemann initial data in terms of classical elementary waves. There are some nonclassical situations [13,23,22] when for few cases of initial data the Riemann problem fails to contain a weak L ∞solution. For that when we solve the Cauchy problem in this nonclassical situation, we have to introduce delta shock type singularities, which are solutions of the system of conservation laws.…”

The limiting behavior of the Riemann solution to the isentropic Euler system for logarithmic equation of state with a source term

“…Recently, Zeidan and Bira studied weak shock waves and its interaction with characteristic shock in polyatomic gas. 31 For more information on delta shock wave, we refer [32][33][34][35][36] and references quoted therein. Shukla and Sharma 37 studied undercompressive shock waves in Hall-magnetohydrodynamics.…”

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

“…In fact, the formation of delta shock wave as well as vacuum state in the Riemann solutions of (1.3) and (1.2) has been extensively investigated by virtue of the vanishing pressure method [16][17][18][19][20][21][22][23][24][25][26][27][28] and the flux function limit method. [29][30][31][32][33] However, it is of great interest to discover that although the limit 𝜀 → 0 of Riemann solution of (1.1) and (1.2) is still a delta shock wave for the case u + < u − , but the propagation speed and strength of delta shock wave in the limiting 𝜀 → 0 situation are obviously different from those of delta shock wave solution of the Riemann problem (1.3) and (1.2). That is to say, the Riemann solutions of (1.1) do not converge to those of (1.3) under the same Riemann initial condition (1.2) when the limit 𝜀 → 0 is taken.…”