A variational formulation of kinematic waves: Solution methods
Introduction
It was shown in part I of this work (Daganzo, 2003, Daganzo, 2005) that kinematic wave (KW) traffic problems with a concave flow-density relation are shortest (least cost) path problems. This paper shows how the theory can be put into practice. As in part I, t and x denote time and space; N(t, x), q = ∂N/∂t and k = −∂N/∂x are the vehicle number, flow and density functions; anda “fundamental diagram” that can vary with t and x.
The function Q was assumed to be differentiable and concave in k in (Daganzo, 2003, Daganzo, 2005), but the results extend trivially to piecewise differentiable functions, possibly discontinuous in t and x. This will be the working assumption now. Concavity implies that the wave velocity w = ∂Q/∂k is non-increasing in k. The density can take any real value, although in well posed traffic problems it automatically remains in a range [0, κ], where q ⩾ 0; see Fig. 1(a). It will also be assumed that Q(0, t, x) = 0, as shown in the figure.1
The main result of part I is that if N has been defined for all points B on a boundary D (i.e., NB is known for all B ∈ D) and a function Q is given, then the value of N at any point P in the solution domain, NP, is given bywhere is a “valid” space-time path x(t) from the boundary to P, is the vehicle number at the beginning of , VP is the set of all valid paths from the boundary to P, and is the “cost” of , defined below. Valid paths are piecewise differentiable curves with slopes x′ in the range of allowable wave-speeds [w(∞, t, x), w(−∞, t, x)]. If we use ΔBP for the minimum of across all valid paths from a generic point B to P, with ΔBP = ∞ if there is no valid path from B to P, then (2) can also be expressed as an ordinary minimization:
The expression for the cost of a valid path is:where tB and tP are the times associated with the path endpoints and R is given by:see Fig. 1. Given our assumptions, it is easy to show that in the range of valid wave velocities, R is non-increasing and convex in x′, and satisfies R⩾0 everywhere, as shown by the figure.2 Definition (4) applies whether or not B ∈ D.
It was explained in part I that R is the maximum rate at which traffic can pass an observer moving with speed x′. Like Q, R has units of vehicles/time and can be defined a priori independently of the boundary data; i.e., it is a property of the highway. Thus, (2), (3), (4), (5) define a continuum shortest path problem from D to the solution domain. It was also explained that if Q is strictly concave shortest paths are waves, and that while this is also true if Q is merely concave, in this case there can also be non-differentiable shortest paths.
These results apply to physically “well-posed” problems, which satisfy the following conditions: (i) the solution domain only includes points P that can be reached by a valid path from the boundary; and (ii) if a valid path issued from B ∈ D ends at B′ ∈ D then . Condition (i) ensures that NP can be calculated; and (ii) that there are no ambiguities. We also assume, that the continuum cost function is uniformly bounded, , as occurs in traffic flow; and that the boundary data satisfy a Lipschitz continuity condition for some α > 0; i.e.:where ∣B − B′∣ is the L1 distance between B and B′: .
The rest of this paper shows how to solve KW problems with discrete networks and evaluates the method’s accuracy. Errors are shown to be uniformly bounded—zero in many important cases. The new method improves on conservation-law procedures because it exploits more fully the properties of Q. Section 2, below, characterizes useful networks. Sections 3 Homogeneous problems, 4 Inhomogeneous problems describe the procedure for homogeneous and inhomogeneous problems.
Section snippets
Networks
This section introduces some definitions and six related facts that indicate when/how a continuum problem can be replaced by a network problem.
Homogeneous problems
Section 3.1 defines relevant network structures, and Section 3.2 how they should be adapted to specific problems. In Section 3.1 and the first two subsections of Section 3.1, Q is piecewise linear; this is relaxed in Section 3.1.3.
Inhomogeneous problems
This section examines four generalizations: (i) discrete shortcuts in a homogeneous continuum; (ii) piecewise homogeneous problems; (iii) self-similar highways; and (iv) general problems.
Acknowledgement
PhD candidate Jorge Laval offered many constructive comments. Research supported by NSF Grant CMS-0313317 to the University of California, Berkeley.
References (5)
A variational formulation of kinematic waves: Basic theory and complex boundary conditions
Transportation Research B
(2005)- Daganzo, C.F., 2003. A variational formulation for a class of first order PDEs. Technical Report UCB-ITS-RR-2003-03....
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