Phase transitions in nonequilibrium traffic theory

This paper uses the center difference scheme of Lax-Friedrichs to numerically solve a newly developed continuum traffic flow theory and the kinematic theory of Lighthill and Whitham, and Richards, and it studies the flow-concentration phase transitions in flow containing both shock and rarefaction waves. A homogeneous road with finite length was modeled by both theories. Numerical simulations show that both theories yield nearly identical results for two representative Riemann problems-one has a shock solution and the other a rarefaction wave solution. Their phase transition curves, however, are different: those derived from the new theory have two branches-one for acceleration flow and one for deceleration flow, whereas those derived from the LWR theory comprise a single curve-the equilibrium curve. The phase transition curves in the shock case agree well with certain experimental observations but disagree with others. This disagreement may be resolved by studying transitions among nonequilibrium states, which awaits further development of a more accurate finite difference approximation of the nonequilibrium theory.