The impact of source terms in the variational representation of traffic flow
- 1. The impact of source terms in the varia2onal representa2on of traffic flow JORGE LAVAL, GUILLAUME COSTESEQUE* AND BARGAVARAMA CHILUKURI TRB, Jan 2016 *Inria Sophia-Antipolis, France
- 2. 2 Mo2va2on • VariaAonal theory (VT) was an important milestone to solve the kinemaAc wave model: – Allows analyAcal global soluAons – No need of shocks and/or entropy condiAons (as in the method of characterisAcs) • Current approximaAon methods for the Macroscopic Fundamental Diagram (MFD) of urban networks rely on VT • èHere we revisit VT now including Eulerian lateral inflows and oulows. 2
- 3. 3 Main result • VT soluAons: – OK in Eulerian coordinates for exogenous source terms – but not when they are a funcAon of traffic condiAons (merge model e.g.) • In discrete Ame source terms become exogenous – à improved numerical soluAon methods • In Lagrangian-Space and Lagrangian-Time coordinates, VT soluAons may not exist even if source terms are exogenous. 3
- 5. 5 Exogenous inflow ΓP u t x tB B=( , )ytBy -w time P = ( , )t x u tt =B 0 By -w time P xU xD space A(ξ)=area ξ( )t all valid paths in this area defineVBP u -w (a) (b)space
- 6. 6 Extended Riemann problems (ERP) u t x0 y -w time P xU xD kU kD u -w 0 upper path middle path lower path aU aD space 6
- 8. 8 Numerical solu2on methods • Idea: in discrete Ame stepping methods, endogenous inflows computed from the previous Ame step become exogenous for the current Ame step and therefore the VT soluAon may be applied. • 2 methods are presented: – Godunov's method (KinemaAc Wave model) – VariaAonal networks (Hamilton-Jacobi model) 8
- 11. 11 Godunov's method - Example k,veh/km time, min CT ruleERP ruleexact solution (eqn 2) (a) q,veh/hr time, min (b) optimalcandidate time, min (c) 10 20 30 40 2 4 6 8 RMSE Δt, sec (d) k=K t=2.35 xU xD x0 y1* 11
- 14. 14 Main result • VT soluAons: – OK in Eulerian coordinates for exogenous source terms – but not when they are a funcAon of traffic condiAons (merge model e.g.) • In discrete Ame source terms become exogenous – à improved numerical soluAon methods • In Lagrangian-Space and Lagrangian-Time coordinates, VT soluAons may not exist even if source terms are exogenous. 14