Formation of delta shock and vacuum state for the pressureless hydrodynamic model under the small disturbance of traffic pressure

The formation of delta shock and vacuum state for the pressureless hydrodynamic model is investigated in detail under the small disturbance of traffic pressure. Exact Riemann solutions for the perturbed system can be constructed explicitly for four different possible structures. Asymptotically, the perturbed Riemann solution involving two shocks will collapse to a single delta shock, and the perturbed Riemann solution involving two rarefaction waves will degenerate into a solution containing two contact discontinuities along with the vacuum state between them when the perturbed parameter goes to zero. It should be stressed here that the internal state in each of the two rarefaction wave-fans turns out to be the vacuum state gradually in such a limiting case, which differs obviously from the previous result that each of the two rarefaction wave-fans is compressed globally to be a single contact discontinuity. Additionally, some typical numerical results exhibiting the formation process of delta shock and vacuum state are presented to verify our theoretic results.