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# Nonlocal reaction traffic flow model with on-off ramps

• * Corresponding author: Luis Miguel Villada
• We present a non-local version of a scalar balance law modeling traffic flow with on-ramps and off-ramps. The source term is used to describe the inflow and output flow over the on-ramp and off-ramps respectively. We approximate the problem using an upwind-type numerical scheme and we provide $\mathbf{L^{\infty}}$ and $\mathbf{BV}$ estimates for the sequence of approximate solutions. Together with a discrete entropy inequality, we also show the well-posedness of the considered class of scalar balance laws. Some numerical simulations illustrate the behaviour of solutions in sample cases.

Mathematics Subject Classification: Primary: 65M08; Secondary: 35L45, 90B20.

 Citation: • • Figure 1.  Illustration of the model setting

Figure 2.  Example 1. Numerical approximations of the problem (2.1). Dynamic of Model 1 vs. Model 2 at (a)$T = 0.5$, (b)$T = 2$, (c)$T = 5$, (d)$T = 7$

Figure 3.  Example 2. Numerical approximations of the problem (2.1) at $T = 5$. Comparison of local and non-local versions of the model (2.1) with $\delta = 0$ and different values for $\eta$

Figure 4.  Example 3. Numerical approximation at time $T = 0.3$. (a) Model 1, Model 2 satisfying a maximum principle and Model 0 not satisfying a maximum principle. (b) Zoom of a part of (a)

Figure 5.  Example 4. Dynamic of the model (2.1). Behavior of the numerical solution computed with Algorithm 3.1 by means of Model 1 and Model 2 at time (a)$T = 1$, (b)$T = 2$, (c)$T = 5$, (d)$T = 7$

Table 1.  Example 2. $\mathbf{L^{1}}$ distance between the approximate solutions to the non-local problem and the local problem for different values of $\eta$ at $T = 5$ with $\Delta x = 1/1000$

 $\eta$ 0.1 0.05 0.01 0.004 $\mathbf{L^{1}}$ distance 0.28 0.16 0.036 0.011
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