Stability of the Riemann solution for a 2 × 2 strictly hyperbolic system of conservation laws

Abstract

In this work, we study the system of conservation laws that is strictly hyperbolic and whose Riemann solution contains delta shock waves as well as classical elementary waves. In order to study stability, we consider the linear approximation of flux functions with three parameters. The approximation does not affect the structure of Riemann solution. Furthermore, we prove that the solution of the Riemann problem for the approximated system converges to the solution of the original system when the perturbation parameter tends to zero.

The adiabatic exponent limits of Riemann solutions for the extended macroscopic production model

“…First of all, the vanishing pressure method was extensively used such as in [8,11,12,22,26,30,33,41]. Subsequently, the approach of the so-called flux function limit was also adopted such as in [29,40,[44][45][46]. Recently, the approach of the adiabatic exponent limit has been well carried out for example in [15, 31, 34-36, 48, 49].…”

Self‐similar viscosity approach to the Riemann problem for a strictly hyperbolic system of conservation laws

“…So we may visualize the distinction between the delta shock wave to the ordinary shock wave as more characteristics enter the delta shock wave than the latter. Diverse methods can be used to explore system of hyperbolic conservation laws in the context of Riemann problem such as vanishing pressure limit method, [14][15][16][17][18] weak asymptotic method, 19 distributional product method, 20,21 flux approximation method, [22][23][24][25] split delta function method, 26,27 etc. In Joseph and Sahoo, 28 vanishing viscosity method is used for a system consisting of four conservation laws.…”

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

“…For example, one may refer to other works [9][10][11][12][13][14][15] about the related concepts of delta shock wave. 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).…”