Establish stochastic generator closure for triangular fundamental diagrams

Establish a stochastic generator and front-intensity closure for the triangular LWR fundamental diagram, including contact-wave transport at affine branches and a well-defined treatment of the nondifferentiable critical-density state, and determine whether smooth strictly concave approximations converge to that closure.

Background

The analytical closure developed in the paper assumes a smooth, strictly concave fundamental diagram, whereas the triangular fundamental diagram used in traffic applications is piecewise linear and nondifferentiable at capacity. The numerical experiments compare smooth approximations with the exact deterministic triangular solution and show decreasing discrepancies over the tested finite horizons, but they do not establish a stochastic closure for the nonsmooth model.

A complete result would need to handle contact waves on affine branches, define the temporal drift at the critical density where the derivative is nonunique, and prove convergence of the evolved generator and front-intensity law as the smoothing parameter tends to zero. The problem is important for extending the framework from smooth theoretical fluxes to the triangular diagrams commonly used in traffic-flow computation.

References

The finite-horizon results support the examined deterministic approximation; they do not establish convergence of _t, F_t, or their common operator core as \varepsilon\to0, or a uniform long-time or long-distance accuracy guarantee. A direct triangular stochastic closure would require a justified treatment of contact transport and the critical state, together with a corresponding closure and interaction balance beyond the smooth proof used here.

— Stochastic Shockwave Dynamics in First-Order Traffic Flow  (2610.23022 - Zhang et al., 16 Oct 2026) in Section 2.2.3, “Triangular diagram”; Section 2.3, “Triangular diagrams, contact waves, and closure scope”; Conclusion