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Heterogeneity in scalar conservation laws: approximation and applications

Abstract : This thesis is devoted to the treatment of heterogeneity in scalar conservation laws. We call heterogeneous a conservation law which is not invariant by space translation. These equations arise for instance in traffic flow dynamics modeling. The presence of traffic lights or roads that have a variable maximum speed limit are examples of mechanisms which lead to heterogeneous conservation laws. Considering such equations is a way to expand macroscopic traffic flow models. We tackle three classes of inhomogeneous problems for which we extend the mathematical framework for both the theoretical analysis and the numerical approximation. We fully investigate the treatment of heterogeneity when one or several moving interfaces are added in the classic LWR model for traffic flow. Flux constraints are attached to each interfaces. The resulting class of models can be used to take into account the presence of slow moving vehicles that reduce the road capacity and thus generates moving bottlenecks for the surrounding traffic flow. They can also describe the regulating effect of autonomous vehicles. In this framework, the interfaces and the constraints are linked in a nonlocal way to the traffic density and/or to an orderliness marker describing the state of the drivers. The description of the heterogeneity caused by the variations in the drivers' organization leads to the analysis of a so-called second order model. The numerical aspect plays a central role in the analysis of these traffic flow models. We construct robust numerical schemes and establish specific techniques to obtain compactness of the approximate solutions. Proving the convergence of these schemes lead to existence results. Finally, with the space-dependent LWR traffic flow model in mind, we theoretically analyze a class of scalar conservation laws with explicit space dependency. Classical results such as well-posedness or the link to the associated Hamilton-Jacobi equation are obtained under a set of assumptions more fitting with the modeling hypothesis. With applications that go beyond traffic modeling in mind, we aim to tackle initial data identification problems.
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Contributor : Abraham Sylla Connect in order to contact the contributor
Submitted on : Wednesday, September 15, 2021 - 2:07:07 PM
Last modification on : Tuesday, October 12, 2021 - 5:20:53 PM


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  • HAL Id : tel-03303049, version 3



Abraham Sylla. Heterogeneity in scalar conservation laws: approximation and applications. Mathematics [math]. Université de Tours, 2021. English. ⟨tel-03303049v3⟩



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