Capturing shock waves in inelastic granular gases

Shock waves in granular gases generated by either a vertically vibrated granular layer or by hitting an obstacle at rest are treated by means of a shock capturing scheme that approximates the Euler equations of granular gas dynamics with an equation of state (EOS), introduced by Goldshtein and Shapiro [ J. Fluid Mech. 282 (1995) 75], that takes into account the inelastic collisions of granules. We include a sink term in the energy balance to account for dissipation of the granular motion by collisional inelasticity, proposed by Haff [J. Fluid Mech. 134 (1983) 401], and the gravity field added as source terms. We have implemented an approximate Riemann solver, due to the second author [J. Comput. Phys. 125 (1996) 42], that works robust under low granular temperatures, high Mach numbers and near close-packed limit, damping post-shock oscillations. We have performed several numerical tests to show numerical evidence of the above features. We have computed the approximate solution to the following problems: a one-dimensional granular gas falling on a plate under the acceleration of gravity until close-packed limit, various one-dimensional blast waves evolving in time in the absence of gravity, a one-dimensional vertically vibrated granular layer under a sinusoidal perturbation and the two-dimensional reflecting shock wave generated when granular gas hits an angular obstacule through the acceleration of gravity.