Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model

Abstract

In this paper, we propose a macroscopic model that describes the influence of a slow moving large vehicle on road traffic. The model consists of a scalar conservation law with a nonlocal constraint on the flux. The constraint level depends on the trajectory of the slower vehicle which is given by an ODE depending on the downstream traffic density. After proving well-posedness, we first build a finite volume scheme and prove its convergence, and then investigate numerically this model by performing a series of tests. In particular, the link with the limit local problem of [M. L. Delle Monache and P. Goatin, J. Differ. Equ. 257 (2014), 4015-4029] is explored numerically.

A Second Order Numerical Scheme for Large-Eddy Simulation of Compressible Flows

“…Invoking (7), ( 3) and (2), we have 1 t J$T if K T Vw ■ nTF = K, f Vw ■ ($-) nTF = K$ f Vw ■ nff J(pf and (26) gives (K4>,fVpK+TVt' Vw)f = (K$,fVVt , Vw)f + y ' (vf -Vt , K$,fVw ■ nfF)f FeFf = (K$f f VpK+lF, Vw)f, the conclusion following fromthedefinition ofpK+ f. Since w is arbitraryin Pk+1 (T), this proves that pk K+-rV.T and pK+ fVf have the same gradient. Using (4) and (24) we also see that they have same average on T, which concludes the proof of (25). □…”

“…F eFj FkjVt > 1)t = (vT, 1)T • (24) Lemma 3 (Transport of potential reconstruction) It holds pK+TvT = pk+1 vT K,A,T T VvT e UT, (25) where vT = (vT, (vF)FeFT ) e UT is the transported vT, and pK+J T ^ the potential reconstruction on T for the diffusion tensor K^t.…”

“…Définition 3 (FLMmethod) Given the initial values (p0,h, m0,h, E0,h ) e Qh x Qh x Qh, we seek a piecewise constant approximation (ph, mh, Eh) e Qh x Qh x Qh which solves at any time t e (0, T] the following equations: By Dt we have denoted (continuous) time derivative; in practical implementation one can use any suitable ODE solver in order to approximate (15). In our recent work [15] we have shown that the FLM method (15) satisfies the following invariant domain-preserving properties, see [23,24] where this notion was firstly introduced.…”

“…Sticko and Kreiss [24] have developed penalty terms for stabilizing the solution of the wave equation, written as a second-order equation. Gürkan and Massing [10] propose a stabilization for solving the steady advection-reaction equation.…”

“…Conversely, if a precipitate is already present and the environmental conditions are such that it can dissolve we will have an increment of the porosity (i.e., void space) and the creation of ions that can be transported by the liquids. Some reference on this subject are: reactive transport on porous media at pore-scale [12,22,30] and at macro-scale [1,14,25] with experiment comparison [24]. For an micro to macro upscaling procedure see [31,32].…”