Flux approximation to the Euler equations of compressible fluid flow for extended Chaplygin gas

The concentration and cavitation are fundamental and physical phenomena in fluid dynamics, which can be mathematically described by delta shock waves and vacuums, respectively. In this paper, we are concerned with the Euler equations of compressible fluid flow when the state equation is governed by the extended Chaplygin gas, an important candidate for describing dark matter and dark energy. Our main objective is to apply the flux-approximation method to rigorously investigate the formation of delta shock waves and vacuums and observe the concentration and cavitation phenomena. First, the Riemann problem of the compressible fluid flow with n + 2 parameters including flux and pressure is solved. Then, two kinds of flux approximation are discussed, that is, both the flux perturbation and pressure tend to zero, or only the flux perturbation vanishes while the extended Chaplygin gas pressure partly tends to the Chaplygin gas pressure. The results indicate that different manners of flux approximation have their respective effects on the formation of delta shock waves. Finally, several numerical results are presented to confirm the theoretical analysis.