Well-Posedness and Convergence of a Finite Volume Method for Conservation Laws on Networks

Abstract

We continue the analysis of nonlinear conservation laws on networks initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101-128] by extending our analysis to a large class of flux functions which must be neither monotone nor convex/concave. We utilize the framework laid down in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101-128] and prove existence and uniqueness within a natural class of entropy solutions via the convergence of an explicit finite volume method. In particular, this leads to the existence of a semigroup of solutions. The theoretical results are supported with numerical experiments including an experimental order of convergence.

From Heterogeneous Microscopic Traffic Flow Models to Macroscopic Models

“…Here the set G is the limit germ and is the main unknown of our problem. We now define the notion of solution for equation (14), following [21,37]. Definition 1.3.…”

“…Following [21,37], and because the germ G is maximal, the last condition in Definition 1.3 is equivalent to the following entropy inequality:…”

“…The situation is completely different for junctions involving at least 3 branches. Indeed, although many works have been devoted to such junctions (see for instance [5,21,26,28,30,37,43]), the problem is still poorly understood and the general picture is far from clear. For instance, if the germ approach of [6] has been recently extended to general junctions in [21,37] (and we strongly use this extension in the paper), there are still few examples of germs which are maximal and complete; one of the outcome of our paper is to describe a new class of such germs (note however that a particular case was previously discussed in [43]).…”

“…Indeed, although many works have been devoted to such junctions (see for instance [5,21,26,28,30,37,43]), the problem is still poorly understood and the general picture is far from clear. For instance, if the germ approach of [6] has been recently extended to general junctions in [21,37] (and we strongly use this extension in the paper), there are still few examples of germs which are maximal and complete; one of the outcome of our paper is to describe a new class of such germs (note however that a particular case was previously discussed in [43]). On the other hand, models involving more than 3 branches seem far richer than the 1:1 set-up: for instance our junction condition (in terms of germs) can be parametrized by a whole family of increasing functions (in contrast with the 1:1 set-up where there is just a single parameter).…”

“…Precisely, we perform a rescaling in space and time, to pass from the mesoscopic scale to the macroscopic one. At the limit of the rescaling, we show rigorous homogenization of the problem and identify the effective junction condition, which belongs to a general class of germs (in the terminology of [6,21,37]). The identification of this germ and of a characteristic subgerm which determines the whole germ, is the first key result of the paper.The second key result of the paper is the construction of a family of correctors whose values at infinity are related to each element of the characteristic subgerm.…”