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Networks & Heterogeneous Media

December 2018 , Volume 13 , Issue 4

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A Godunov type scheme for a class of LWR traffic flow models with non-local flux
Jan Friedrich, Oliver Kolb and Simone Göttlich
2018, 13(4): 531-547 doi: 10.3934/nhm.2018024 +[Abstract](335) +[HTML](165) +[PDF](497.72KB)
Abstract:

We present a Godunov type numerical scheme for a class of scalar conservation laws with non-local flux arising for example in traffic flow models. The proposed scheme delivers more accurate solutions than the widely used Lax-Friedrichs type scheme. In contrast to other approaches, we consider a non-local mean velocity instead of a mean density and provide \begin{document}$L^∞$\end{document} and bounded variation 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 conservation laws. The better accuracy of the Godunov type scheme in comparison to Lax-Friedrichs is proved by a variety of numerical examples.

Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect
Linglong Du
2018, 13(4): 549-565 doi: 10.3934/nhm.2018025 +[Abstract](188) +[HTML](90) +[PDF](481.77KB)
Abstract:

In this paper, we investigate the existence and long time behavior of the solution for the nonlinear visco-elastic damped wave equation in \begin{document}$\mathbb{R}^n_+$\end{document}, provided that the initial data is sufficiently small. It is shown that for the long time, one can use the convected heat kernel to describe the hyperbolic wave transport structure and damped diffusive mechanism. The Green's function for the linear initial boundary value problem can be described in terms of the fundamental solution (for the full space problem) and reflected fundamental solution coupled with the boundary operator. Using the Duhamel's principle, we get the \begin{document}$ L^p $\end{document} decaying rate for the nonlinear solution \begin{document}$\partial_{{\bf x}}^{\alpha}u$\end{document} for \begin{document}$|\alpha|\le 1$\end{document}.

Influence prediction for continuous-time information propagation on networks
Shui-Nee Chow, Xiaojing Ye, Hongyuan Zha and Haomin Zhou
2018, 13(4): 567-583 doi: 10.3934/nhm.2018026 +[Abstract](170) +[HTML](92) +[PDF](672.08KB)
Abstract:

We consider the problem of predicting the time evolution of influence, defined by the expected number of activated (infected) nodes, given a set of initially activated nodes on a propagation network. To address the significant computational challenges of this problem on large heterogeneous networks, we establish a system of differential equations governing the dynamics of probability mass functions on the state graph where each node lumps a number of activation states of the network, which can be considered as an analogue to the Fokker-Planck equation in continuous space. We provides several methods to estimate the system parameters which depend on the identities of the initially active nodes, the network topology, and the activation rates etc. The influence is then estimated by the solution of such a system of differential equations. Dependency of the prediction error on the parameter estimation is established. This approach gives rise to a class of novel and scalable algorithms that work effectively for large-scale and dense networks. Numerical results are provided to show the very promising performance in terms of prediction accuracy and computational efficiency of this approach.

On boundary optimal control problem for an arterial system: First-order optimality conditions
Ciro D'Apice, Olha P. Kupenko and Rosanna Manzo
2018, 13(4): 585-607 doi: 10.3934/nhm.2018027 +[Abstract](99) +[HTML](45) +[PDF](466.33KB)
Abstract:

We discuss a control constrained boundary optimal control problem for the Boussinesq-type system arising in the study of the dynamics of an arterial network. We suppose that the control object is described by an initial-boundary value problem for \begin{document}$ 1D $\end{document} system of pseudo-parabolic nonlinear equations with an unbounded coefficient in the principle part and the Robin-type of boundary conditions. The main question we study in this part of the paper is about the existence of optimal solutions and first-order optimality conditions.

Effective interface conditions for processes through thin heterogeneous layers with nonlinear transmission at the microscopic bulk-layer interface
Markus Gahn, Maria Neuss-Radu and Peter Knabner
2018, 13(4): 609-640 doi: 10.3934/nhm.2018028 +[Abstract](69) +[HTML](66) +[PDF](626.09KB)
Abstract:

In this paper, we consider a system of reaction-diffusion equations in a domain consisting of two bulk regions separated by a thin layer with thickness of order $ε$ and a periodic heterogeneous structure. The equations inside the layer depend on $ε$ and the diffusivity inside the layer on an additional parameter $γ ∈ [-1, 1]$. On the bulk-layer interface, we assume a nonlinear Neumann-transmission condition depending on the solutions on both sides of the interface. For $\epsilon \to0 $, when the thin layer reduces to an interface $Σ$ between two bulk domains, we rigorously derive macroscopic models with effective conditions across the interface $Σ$. The crucial part is to pass to the limit in the nonlinear terms, especially for the traces on the interface between the different compartments. For this purpose, we use the method of two-scale convergence for thin heterogeneous layers, and a Kolmogorov-type compactness result for Banach valued functions, applied to the unfolded sequence in the thin layer.

Optimal model switching for gas flow in pipe networks
Fabian Rüffler, Volker Mehrmann and Falk M. Hante
2018, 13(4): 641-661 doi: 10.3934/nhm.2018029 +[Abstract](71) +[HTML](49) +[PDF](589.29KB)
Abstract:

We consider model adaptivity for gas flow in pipeline networks. For each instant in time and for each pipe in the network a model for the gas flow is to be selected from a hierarchy of models in order to maximize a performance index that balances model accuracy and computational cost for a simulation of the entire network. This combinatorial problem involving partial differential equations is posed as an optimal switching control problem for abstract semilinear evolutions. We provide a theoretical and numerical framework for solving this problem using a two stage gradient descent approach based on switching time and mode insertion gradients. A numerical study demonstrates the practicability of the approach.

Fluvial to torrential phase transition in open canals
Maya Briani and Benedetto Piccoli
2018, 13(4): 663-690 doi: 10.3934/nhm.2018030 +[Abstract](106) +[HTML](40) +[PDF](1320.7KB)
Abstract:

Network flows and specifically water flow in open canals can be modeled bysystems of balance laws defined ongraphs.The shallow water or Saint-Venant system of balance laws is one of the most used modeland present two phases: fluvial or sub-critical and torrential or super-critical.Phase transitions may occur within the same canal but transitions relatedto networks are less investigated.In this paper we provide a complete characterization of possible phase transitionsfor a case study of a simple scenariowith two canals and one junction.However, our analysis allows the study of more complicate networks.Moreover, we provide some numerical simulations to show the theory at work.

Stability implies constancy for fully autonomous reaction-diffusion-equations on finite metric graphs
Joachim von Below and José A. Lubary
2018, 13(4): 691-717 doi: 10.3934/nhm.2018031 +[Abstract](72) +[HTML](36) +[PDF](534.93KB)
Abstract:

We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph \begin{document}$ G$\end{document} under continuity and Kirchhoff flow transition conditions at the vertices.

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