Singularities in kinematic wave theory: Solution properties, extended methods and duality revisited

•Kinematic wave theory is extended to allow for the singularities arising in car-following models.•The only new feature of the singular solutions is a new form of shock, called a supershock.•The physical properties and laws of evolution of these supershocks are identified.•Extended solution methods for problems with general, not-necessarily concave, FDs are presented.•The e-methods satisfy the duality principle and can be applied in Euler and Lagrange coordinates. According to Euler–Lagrange duality principle of kinematic wave (KW) theory any well-posed initial value traffic flow problem can be solved with the same methods either on the time–space (Euler) plane or the time vs vehicle number (Lagrange) plane. To achieve this symmetry the model parameters and the boundary data need to be expressed in a form appropriate for each plane. It turns out, however, that when boundary data that are bounded in one plane are transformed for the other, singular points with infinite density sometimes arise. Duality theory indicates that solutions to these problems must exist and be unique. Therefore, these solutions should be characterized. The paper shows that the only added feature of these solutions is a new type of shock that can contain mass and we call a supershock. Nothing else is different. The evolution laws of these shocks are described. Solution methods based on these laws for problems with singularities are also presented. The methods apply to problems with monotone speed–density relations and not necessarily concave fundamental diagrams. In accordance with duality theory they can be used with both, the Euler and Lagrange versions of a problem.