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Start of funding 01.07.2017

Nonlocal Balance Laws – Mathematics with application in traffic flow and chemical engineering

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On the one hand, traffic flow models are used to improve the control of traffic flows in real world scenarios e.g. with the aim to reduce congestion. On the other hand, the quality of numerous pharmaceutical products and the efficiency of their synthesis processes are significantly determined by the ripening behavior of particles during the procedure. Because of the far-reaching consequences, our research project aims to a detailed mathematical description of these phenomena. Thereby, it can be shown that both ostensibly completely different applications can be modeled by so-called nonlocal balance laws (NBLs). Based on the mathematical theory for NBLs we will derive schemes for solving these equations numerically and study their accuracy for a wide range of non-smooth initial datas and velocity functions. We also aim to compute sensitivities of the model with respect to process conditions to deduce necessary first order optimality conditions for optimal control problems relying on NBLs. Finally, an extension of the previously mentioned theory, which strongly relies on fixed-point arguments and characteristics, to the case of multi-dimensional spatial variables and on systems of NBLs is desired.

Final report:
During the funding period 2017/2018 granted by BaCaTeC, we have proven and generalized results on nonlocal balance laws. The generality of our approach enables the research community to use the studied models in a rigorous mathematical framework and guarantees convergence of several corresponding numerical algorithms. These classes of algorithms for solving nonlocal balance laws based on characteristics will be published to the end of the current funding period. The basic idea which is shared by all of these numerical methods is the fact that we, at first, consider a piecewise constant spatial discretization of the solution of the nonlocal balance law. Afterwards, a time integration scheme can be used to solve the therein derived characteristic ODEs numerically. This method allows us to obtain up to linear convergence of the numerical solution to the analytical one for which a detailed solution theory was established in [1]. A further advantage of our algorithms is that they are nondissipative due to the fact that we do not consider a fixed but a flexible grid moving along the characteristic lines which results in a more precise numerical solution without any numerical diffusion.

The quality of the conducted research can easily be seen by the highly ranked journals where the results have been published in (e.g. Journal of Differential Equations and Journal of Mathematical Analysis and Applications). Beside this, the developed algorithms play a key role in optimizing chemical ripening processes (collaboration of L. Pflug, G. Leugering with Prof. W. Peukert, Institute of Particle Technology/FAU Erlangen-Nürnberg) and have already proven to be superior to existent algorithms resulting in more precise simulations even with lower resolution.

We also increased the applicability of our models to real world phenomena within traffic research since we elaborated in [4] a rigorous mathematical theory for nonlocal balance laws on bounded domains which models traffic flow on a single road nonlocally. Again, we strongly rely on the approved ansatz of the careful discussion of a fixed-point equation in the nonlocal term and the characteristic ODE system.

The mathematical framework for the optimization of traffic flow is significantly more complex and further work has to be carried out to contribute to the modeling of congestion and traffic jams in the real world, which then can be used to avoid or decrease their impact. However, within [2,3] we did not only provide an exhaustive solution theory for the multi-dimensional (in the spatial variable) and system case – which is, for instance, relevant for the analysis of crowd dynamics, particle size distributions depending on several disperse properties and multi-commodity traffic problems on networks – but we were also able to build a solid basis for its mathematical optimization since, among others, they contain stability analysis with respect to input data which is crucial for the task of finding optimal controls of optimal control problems governed by nonlocal balance laws.

The research cooperation between the participants has strongly benefited from the BaCaTeC travel funding: M. Spinola’s research stay in Feb. 2018 at UC Berkeley and the following research stay of A. Keimer in Feb. 2018 at FAU Erlangen-Nürnberg has laid the foundation for mutual research activities instantiated and detailed in [4,5]. The current research stay of L. Pflug at UC Berkeley (Sep./Oct. 2018) resulted in [6] and has shown the necessity to dive deeper into nonlocal balance laws modeling traffic flow on road networks. It has also led to the outcome that nonlocal modeling can be interpreted as a generalization of the well-known and rich class of local models when carrying out a limit process in the nonlocal term.

The current funding period has raised many new challenging and highly applicable mathematical questions, which we will address in the future. For instance, nonlocal balance laws have drawn even more attention but lack for now the capability to be generalized to networks of traffic, multi-lane (mandatory lane changes such as on ramping and nonmandatory ones such as in free flow) and multi-class traffic problems. The respective nonlocal formulations – as a generalization/extension due to the usage of the in [4] defined external impact of the outflow – can be supplemented by real time navigational apps (Google Waze, Google Maps, Apple Maps, etc.) and can improve the design of routing controllers as e.g. traffic lights and speed limits.

Finally, as the considered class of models is quite general, they still entail applications in chemical engineering, supply chains and opinion formation models as had been described in the application of the current funding period.

Publications:
1. Keimer, A. and Pflug, L., “Existence, uniqueness and regularity results on nonlocal balance laws”, Journal of Differential Equations (2017)
2. Keimer, A., Pflug, L. and Spinola. M., “Existence, uniqueness and regularity of multi-dimensional nonlocal balance laws with damping”, Journal of Mathematical Analysis and Applications (2018)
3. Keimer, A., Leugering, G. and Sarkar, T., “Analysis of a system of nonlocal balance laws with weighted work in progress”, Journal of Hyperbolic Differential Equations (2018)
4. Keimer, A., Pflug, L. and Spinola, M., “Nonlocal scalar conservation laws on bounded domains and applications in traffic flow”, SIAM Journal of Mathematical Analysis and Application (accepted, 10/18)
5. Bayen, A., Keimer, A., Porter, E. and Spinola, M., “Time-continuous instantaneous and past time routing on traffic networks: A mathematical analysis on the basis of the link delay model”, submitted to Annual Review of Control, Robotics, and Autonomous Systems (10/18)
6.Keimer, A. and Pflug, L., “On approximation of local conservation laws by nonlocal conservation laws”, submitted to SIAM Journal of Applied Dynamical Systems (10/18)