A conservative analytical solution of the shock–contact–shock Riemann problem for dilute granular gas

In the work of Fouda (2021), an approximate analytical solution was derived for the shock–contact–shock (SCS) Riemann problem for dilute granular gas with slightly inelastic particle collisions. The one-dimensional Euler equations were expanded in the power series of the inelastic dissipation parameter using a regular perturbation method. The first order system of this expansion was solved using characteristic decomposition of the hydrodynamic variables without taking into account the jump and conservation conditions across the three discontinuous lines. In the present paper, an improved conservative solution of this first order system is obtained by incorporating the Rankine–Hugoniot (RH) jump conditions across the shocks, and the conditions of pressure and velocity continuity across the contact, resulting in a full first order accuracy of the approximate analytical solution. It is shown that the first order jumps across the three discontinuities and their propagation speeds are linear functions of time, with additional nine variables that have been evaluated. The predictions of the improved conservative approximate analytical solution showed better agreement with the numerical solution using a shock capturing scheme, which allows the extension of this solution method to other hydrodynamic systems of balance laws.