Essence of conservation forms in the traveling wave solutions of higher-order traffic flow models

This paper shows the essence of conservation forms when applying the weak solution theory to solve the traveling wave solution of a wide cluster in the Payne-Whitham (PW) model. The consideration of the conservation form for the acceleration equation is an important ingredient in the development of higher-order traffic flow models, but it is largely ignored in the research community. To fix the idea, we define two conservation forms for the same PW model, and consequently derive two solutions with different sets of characteristic parameters of the wide cluster. The analytical results are in good agreement with those that are obtained from numerical simulations. Moreover, these two solutions are also shown to be asymptotic to those of the well-known Kühne and Kerner-Konhaüser models with a viscosity term. More importantly, the careful treatment of the conservation form for the acceleration equation closes the important gap in the literature. Without the conservation form, the solution obtained depends very much on the design of numerical schemes, and can be quite arbitrary and may not adequately conform to the physically relevant properties.