Thesis Summary:

This thesis aims at the modeling, analysis and numerical approximation by means of finite volume methods, of spatially one-dimensional hyperbolic balance laws, with nonlocal flux function, motivated by applications in sedimentation and vehicular traffic. Particularly, we are interested in studying well-posedness and to design efficient numerical schemes to compute approximate solutions of new proposed models in the frame of the target applications. First we intend to model a batch sedimentation process in a closed column, for it we consider an initial boundary values problem (IBVP) for a nonlocal conservation law in which the nonlocal term is given by the convolution between a kernel function and the velocity of sedimentation. This nonlocal operator is assumed to be aware of boundary terms. For this first proposed model we study well-posedness and adapt a nonlocal version of a Hilliges-Weidlich (HW)-type numerical scheme. Specifically, it is proved that the uniqueness of entropy weak solutions to nonlocal model follows from the Lipschitz continuous dependence of the solution on initial and boundary data; likewise, by means of the numerical scheme we provide compactness estimates along with a discrete entropy solution, which show the existence of weak solutions and the convergence of the sequence of approximate solutions to an entropy weak solution of the nonlocal problem. We compare the HW-type scheme with schemes based on the Lax-Friedrichs flux through numerical examples. A second-order HW-type scheme based on MUSCL methods is also presented. Second, we model the traffic dynamics on a road with rough conditions, by means of a conservation law whose nonlocal flux has a hidrance term and a single spatial discontinuity. The nonlocal term reflects that drivers adapt their velocity with respect to what happens in front of them. These assumptions lead to a expression flux in which the velocity depends on a convolution between density of vehicles and a function kernel. We approximate the problem by means of the HW-type numerical scheme proposed above and provide some uniform estimates on the sequence of approximate solutions what allow us to prove existence of a entropy weak solution. We also provide L1 stability and therefore uniqueness of entropy solutions. Subsequently, we introduce a model that describes the vehicular traffic dynamics on a road with on- and off ramps, for which we consider a nonlocal balance law in which the source term independently describes the inflow and outflow via on-ramp and off-ramps. The source term depends on a downstream convolution term that describes that drivers on the on-ramp can see what happens behind and in front of them on the main road. Existence of entropy weak solutions is proved approximating the numerical solutions by means of the HW-type scheme along with an operator splitting to account the reaction term and providing L∞ and BV estimates to sequence of approximate solutions. Uniqueness of entropy weak solution is proved by means L1−Lipschitz continuous dependence of solution on initial datum, on-ramp rate and off-ramp rate. We also study numerically the limit model as support of the kernel function tends to zero and also are presented some numerical simulations illustrating the dynamic of the studied model. Then, motivated by optimization and control problems, we study the dependence of solutions to the vehicular traffic model with ramps introduced above on the convolution kernel given in the source term. We obtain an estimate of the dependence of the solution with respect to the kernel function in the source term, the initial datum, on-ramp rate and off-ramp rate. Stability is obtained from the entropy condition through doubling of variable technique. We also provide some numerical simulations illustrating the dependencies above for some cost functionals. Finally, in order to model vehicular traffic flow on a two-lane and two-way road where drivers have a preferred lane, the lane on their right, and the left one is used only for overtaking slower vehicles, we propose a system of nonlocal balance laws. In this model the convective part describes the intra-lane dynamics, for this reason the flux functions consider local and nonlocal terms, namely, the velocity function in each lane depends locally on the density of vehicles of the preferential class and on a nonlocal form on the density of vehicles of the another class coming in opposite direction on the same lane overtaking; in turn, the source terms describe the inter-lane coupling between the two lanes, so that we consider the overtake and return criteria dependent on a weighted mean of the downstream traffic density of preferred class and a weighted mean of downstream traffic density of the classes traveling in opposite direction. We approximate the solutions of the problem by means of the HW-type numerical scheme developed in this thesis and prove existence of weak solutions by means of compactness estimates. We also show some numerical simulations that describe the behavior of the solutions in different situations. Key Words: Nonlocal conservation laws, nonlocal balance laws, initial boundary values problem, convolution term, kernel functions, entropy weak solution, HW type numerical scheme, discontinuous flux function, macroscopic vehicular traffic models, Lighthill-Whitham-Richards traffic model, on- and off-ramps, wellposedness, multilane traffic model, two way and two lane traffic model.

ISI Publications from the Thesis

Felisia A. CHIARELLO, Harold D. CONTRERAS, Luis M. VILLADA: Existence of entropy weak solutions for 1D non-local traffic models with space-discontinous flux. Journal of Engineering Mathematics, vol. 141, no. 9, (2023).

Raimund BüRGER, Harold D. CONTRERAS, Luis M. VILLADA: A Hilliges-Weidlich-type scheme for a one-dimensional scalar conservation law with nonlocal flux. Networks and Heterogeneous Media, vol. 18, pp. 664-693, (2023).

Felisia A. CHIARELLO, Harold D. CONTRERAS, Luis M. VILLADA: Nonlocal reaction traffic flow model with on-off ramps. Networks and Heterogeneous Media, vol. 17, no. 2, pp. 203-226, (2022).