A variational formulation for higher order macroscopic traffic flow models: Numerical investigation

• We consider a generic class of macroscopic traffic flow models on an homogeneous section.
• We propose a numerical methodology to solve this family of models known as GSOM.
• The model is recast under its Lagrangian form for more efficiency.
• The methodology uses the Lax–Hopf formula for Hamilton–Jacobi problems.
• We investigate data assimilation problems, for initial and boundary conditions.

This paper deals with numerical methods providing semi-analytic solutions to a wide class of macroscopic traffic flow models for piecewise affine initial and boundary conditions. In a very recent paper, a variational principle has been proved for models of the Generic Second Order Modeling (GSOM) family, yielding an adequate framework for effective numerical methods. Any model of the GSOM family can be recast into its Lagrangian form as a Hamilton–Jacobi equation (HJ) for which the solution is interpreted as the position of vehicles. This solution can be computed thanks to Lax–Hopf like formulas and a generalization of the inf-morphism property. The efficiency of this computational method is illustrated through a numerical example and finally a discussion about future developments is provided.