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On a diffusively corrected kinematic-wave traffic flow model with changing road surface conditions. (English) Zbl 1055.35071
The authors of this very interesting paper study a generalization of the Lighthill-Whitham-Richards kinetic wave traffic flow model for unidirectional flow on a single-lane highway. This model includes the following principle of conservation of cars: $$\partial_t\rho (x,t)+\partial_x(\rho (x,t)v (x,t))=0$$, where $$\rho$$ is the density of cars as a function of distance $$x$$ and time $$t$$ and $$v (x,t)$$ is the velocity of the car located at point $$x$$ at time $$t$$. The decisive constitutive assumption is that $$v$$ is a function of the density $$\rho$$ only, which results in the scale conservation law $$\partial_t\rho +\partial_x(\rho v (\rho ))=0,$$ $$x\in \mathbb{R}$$, $$t >0,$$ where $$v(\rho )$$ is the car velocity at density $$\rho$$. Here the authors apply a very important idea, that the driver of each car instantaneously adjusts his velocity to the local car density. Having in mind this as well as the influence of both abruptly changing road surface conditions and drivers’ reaction time and anticipation length, the authors propose two extended models written in the form $$\partial_t\rho +\partial_x f(\rho )=\partial_x^2D(\rho )$$, $$x\in \mathbb{R}$$, $$t>0$$, where $$D$$ is a certain function. Moreover, this model is modified as $$\partial_t\rho +\partial_x (\gamma^1(x) f(\rho ))=\partial_x(\gamma^2(x)\partial_xR(\rho ,\gamma^2(x)))$$, where $$\gamma^1$$, $$\gamma^2$$, $$R$$ are certain nonnegative functions. The result is a strongly degenerate convection-diffusion equation. The diffusion term, accounting for the drivers’ behavior, is effective only where the local car density exceeds a critical value, and the convective flux function depends discontinuously on the location. It is shown that the validity of the considered models is supported by a recent mathematical well-posedness (existence and uniqueness) theory for quasilinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Finally, the authors give eight numerical examples illustrating the considered theory.

##### MSC:
 35L65 Hyperbolic conservation laws 35K65 Degenerate parabolic equations 90B20 Traffic problems in operations research
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