Exact and grid-free solutions to the Lighthill–Whitham– Richards traffic flow model with bounded acceleration Christian Claudel Assistant professor, Civil, Architectural and Environmental Engineering University of Texas, Austin

Outline Primitive flow models based on LWR - The LWR PDE - Integral formulation and physical interpretation Fast solution methods for LWR - Lax-Hopf formula - Inf-morphism property - Applications Hybrid LWR-bounded acceleration models - Definition - Grid-free computational methods Conclusion

- First derived by Lighthill-Whitham (1955), and extended by Richards (1956) - First order scalar hyperbolic conservation law - Based on the conservation of vehicles, and on the existence of a relationship between flow and density: q=ψ(ρ). ψ(.) is assumed to be concave [Newell 93], [Daganzo 03,06] The LWR PDE Flow ψ(ρ) (veh/h) Density ρ (veh/mile) Source: PeMS

Equivalently, we can define M(t,x) such that: The function M(t,x) is called Moskowitz function. Its spatial derivative is the opposite of the density function; its temporal derivative is the flow function. M(t,x) satisfies the following Hamilton-Jacobi PDE: [Newell 93], [Daganzo 03,06] Integral formulation

M(t,x) can be interpreted as a vehicle label at location x and time t (assuming that no vehicles pass each other). M(t,x) is also known as the cumulative vehicle number in the traffic flow community. [Newell 93], [Daganzo 03,06] vehicle label position Physical interpretation of the Moskowitz function

Outline Primitive flow models based on LWR - The LWR PDE - Integral formulation and physical interpretation Fast solution methods for LWR - Lax-Hopf formula - Inf-morphism property - Applications Hybrid LWR-bounded acceleration models - Definition - Grid-free computational methods Conclusion

The Moskowitz formulation is particularly suited to Lagrangian problems: when no passing is allowed, vehicles keep their labels, which enables the definition of internal conditions (trajectory constraints that apply to the solution) Defining internal conditions for the LWR PDE is possible, but not as easy. [Newell 93], [Daganzo 03,06] Value conditions

Lax-Hopf formula For a boundary data function c(.,.), the solution M c (.,.) is given by: where is the convex transform of ψ [Lax 1973] [Aubin Bayen Saint Pierre SIAM SICON 2009] [Claudel Bayen IEEE TAC part II 2010] Time Position c(.,.)

Inf-morphism property Let us assume that the boundary data c is the minimum of a finite number of lower semicontinuous functions: The solution associated with the above boundary data function can be decomposed as: Inf-morphism [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] Time Position c 12 (.,.) c 1 (.,.) c 2 (.,.) c 3 (.,.) c 8 (.,.) c 5 (.,.) c 6 (.,.) c 4 (.,.) c 7 (.,.) c 9 (.,.) c 10 (.,.) c 11 (.,.)

Physical interpretation [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] Inf-morphism property: The solution associated with a set of boundary conditions is the minimum of the solutions associated with each boundary condition. [Newell 93] [Daganzo 06] [Aubin Bayen Saint-Pierre 07]

Physical interpretation [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] Inf-morphism property: The solution associated with a set of boundary conditions is the minimum of the solutions associated with each boundary condition. [Newell 93] [Daganzo 06] [Aubin Bayen Saint-Pierre 07]

Inf-morphism property Let us assume that the boundary data c is the minimum of a finite number of lower semicontinuous functions: The solution associated with the above boundary data function can be decomposed as: Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] Time Position c 2 (.,.) c 3 (.,.) c 8 (.,.) c 4 (.,.) c 10 (.,.)

Inf-morphism property The solution associated with the above boundary data function can be decomposed as: If is linear, then can be explicitly computed (solution to a 1D convex optimization problem. Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] Position c 12 (.,.) c 1 (.,.) c 2 (.,.) c 3 (.,.) c 8 (.,.) c 5 (.,.) c 6 (.,.) c 4 (.,.) c 7 (.,.) c 9 (.,.) c 10 (.,.) c 11 (.,.)

Semi-analytic formulation Most of the time, we consider piecewise constant initial and boundary conditions for the LWR PDE. This translates into piecewise affine initial and boundary conditions for the HJ PDE. [Daganzo 2005 TR-B] [Daganzo 2006] [Mazare Dehwah Claudel Bayen 2012]

The solution associated with the above boundary data function can be decomposed as: Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] position time

The solution associated with the above boundary data function can be decomposed as: Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] position time

The solution associated with the above boundary data function can be decomposed as: Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] position time

The solution associated with the above boundary data function can be decomposed as: Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] position time

The solution associated with the above boundary data function can be decomposed as: Semi-analytic computational methods [Claudel Bayen IEEE TAC part II 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012] position time

Solutions to affine value conditions [Daganzo 2005 TR-B] [Daganzo 2006] [Claudel Bayen TAC 2010]

Solutions to affine value conditions Example of solution: affine initial condition Lax-Hopf formula: Objective function: convex (in u): constraints also convex (intersection of intervals) : 1-D convex optimization. [Claudel Bayen TAC 2010]

Solution structure Solution structure: [Claudel Bayen TAC 2010]

Extension for solutions to the LWR PDE The solutions to the LWR PDE (density) can also be computed semi analytically: Godunov Lax-Hopf

Pros –Very fast (speed depends upon the problem, and number of points used) –Exact derivation of both the density and Moskowitz function (for the same cost), for any concave fundamental diagram, on any grid –Can integrate probe data Cons –Slower than the CTM if the density has to be computed everywhere Semi-analytic computational methods

Faster algorithm for triangular fundamental diagram

Semi-analytic computational methods Many existing computational methods: For LWR: -Godunov scheme (or equivalently CTM) -Wave-front tracking -Other finite difference schemes (ENO, WENO) For HJ: -Lax Friedrichs schemes (or other numerical schemes) -Variational method (dynamic programming) -Semi-analytic method (for homogeneous problems), which can be used for both HJ and LWR [Daganzo06] [Claudel Bayen IEEE TAC part I & 2, 2010] [Mazare Dehwah Claudel Bayen, TR-B 2012]

Outline Primitive flow models based on LWR - The LWR PDE - Integral formulation and physical interpretation Fast solution methods for LWR - Lax-Hopf formula - Inf-morphism property - Applications Hybrid LWR-bounded acceleration models - Definition - Grid-free computational methods Conclusion

Hybrid models We consider a hybrid model, in which vehicles either follow the LWR model or have a constant acceleration The velocity of the flow of vehicles is defined by Since N is not differentiable, v is not always defined [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Hybrid models We consider a hybrid model, in which vehicles either follow the LWR model or have a constant acceleration The velocity of the flow of vehicles is defined by Since N is not differentiable, v is not always defined [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Modified solution components Let us define “modified” solution components: where [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Modified solution components The modified solution components either satisfy the LWR model, or describe vehicles with constant acceleration They also satisfy an inf-property: [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Modified solution components The modified solution components either satisfy the LWR model, or describe vehicles with constant acceleration They also satisfy an inf-property: [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Analytical derivation of modified solution components Since the solutions components M ci (.,.) are known analytically, we can similarly compute the modified components analytically for initial, upstream, downstream and internal conditions (fixed or moving bottlenecks) Solution structure (initial condition) [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Analytical derivation of modified solution components Solution structure (upstream and downstream boundary condition) [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Analytical derivation of modified solution components Solution structure (internal condition) [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Analytical derivation of modified solution components Solution structure (internal condition) [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Analytical derivation of modified solution components Solution structure (internal condition) [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Analytical derivation of modified solution components Solution structure (internal condition) [Lebacque, 2002] [Leclercq, 2007] [Qiu Abdellatif Abdelaziz Claudel 2013]

Example (I): solution to LWR-bounded acceleration model with a moving bottleneck Applications

Example (II): solution to LWR-bounded acceleration model with a fixed bottleneck (~1 s computational time on a desktop computer) Applications

Conclusion Fast and exact algorithm for computing solutions to the LWR traffic flow model with bounded acceleration Theory works only for specific fundamental diagrams (linear in congestion) Computational cost is identical to the Lax-Hopf algorithm