Two-Phase Fluids Model for Freeway Traffic Flow and Its Application to Simulate Evolution of Solitons in Traffic

A two-phase fluids model for mixing traffic flow on freeways has been proposed, where vehicles are decomposed into two parts (i.e., phases)—slow moving and fast moving–denoted by subscripts $i=1$ and 2, respectively. Based on the fact that both phases should be at rest under traffic jam conditions, it is assumed that the vehicular speeds for both phases are functions of the global traffic density, so that traffic flux can be expressed explicitly, considering that the speed of the second phase may be decreased when the mass fraction of the first phase becomes large. In addition to the relation to global density of traffic, it is assumed that the speed of vehicles in the second phase also depends on the mass fraction for the first phase. By neglecting the traffic generation rate, the governing equations from the mass conservation law were solved numerically with the Yee-Roe-Davis second-order symmetrical total variable diminishing algorithm. Two cases were considered: first, that there exists a soliton in the initial distribution of global density; second, that there is an initial uniform global density. Both cases were allowed to have a soliton in the initial density for the slowly moving vehicles. The numerical results indicate that the evolution of a soliton in traffic is quite different from those in water wave problems. The initial solitary-wave perturbation has been distorted dramatically. It was found that the presence of a soliton just in the slowly moving vehicles can increase global density, which means that traffic mixing can be viewed as a source of density wave production. Under very congested traffic flow, the speed for moving vehicles fast approaches that of vehicles in the first part.