The limits of Riemann solutions to the simplified pressureless Euler system with flux approximation

Abstract

The formation of vacuum state and delta shock wave in the solutions to the Riemann problem for the simplified pressureless Euler system is considered under the linear approximations of flux functions. The method is to perturb the non-strictly hyperbolic system into a nearby strictly hyperbolic system by introducing appropriately the linear approximations of flux functions. The solutions to the Riemann problem for the approximated system can be constructed explicitly and then the formation of vacuum state and delta shock wave can be observed by taking the perturbation parameter tend to zero in the solutions. KEYWORDS delta shock wave, linear flux approximation, pressureless Euler system, Riemann problem, vacuum state

Flux-Approximation Limits of Solutions to the Brio System with Two Independent Parameters

“…This point is the same as that in [20,21]. While when only one parameter ǫ 1 decreases, the strict hyperbolicity of the limiting system is preserved(see Section 6), but the delta shock wave still occurs, which is different from that in [20,21]. In addition, the above results also indicate the fact that different flux approximations have their respective effects on the formation of delta shock waves.…”

“…The main idea of it is to introduce some small perturbed parameters in the flux function of the system, then the limits of solutions to the perturbed system can be studied by taking the perturbed parameters go to zero. This method has been successfully applied to study the formation of delta shock waves, say, Yang and Liu [25] for the nonisentropic fluid flows, Yang and Zhang [26,27] for the relativistic Euler equations, Sun [21] for the transport equations, etc.…”

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

“…Physically, a reasonable perturbation can be used to govern some dynamical behaviours of fluids; hence the flux perturbation problem, which plays an important role in the theory, computation and applications, is worth studying. Using flux approximation, Sun [17] studied the limits of Riemann solutions of simplified pressureless Euler system. The flux function approximation for a system of pressureless gas dynamics with two parameters has been discussed by Shen [18].…”

The asymptotic limits of Riemann solutions for the isentropic drift‐flux model of compressible two‐phase flows

“…18 Recently, it has also been used to investigate the formation of delta shockwave for various Chaplygin gas dynamics systems, see previous studies [19][20][21][22][23][24] for examples. In addition, the method of flux function approximations was proposed firstly in Yang and Liu 25 to study the formation of delta shockwave and vacuum state for the zero-pressure gas dynamical system, which has also been developed in prcvious studies [26][27][28][29][30] for the other systems of hyperbolic conservation laws recently.…”