engleski

Numerical Methods for Hyperbolic Nets and Networks

Hyperbolic nets and networks arise in a broad spectrum of problems. Hyperbolic nets represent structures consisting of slender components, naturally embedded in 3D, whose deformation varies in time via propagating waves. Examples include bridges, carbon nanotubes, or endovascular prosthesis called stents. Hyperbolic networks usually represent fixed structures, on which the flow of some (conserved) quantity is present along branches. Examples include data networks, arterial networks, water channel networks, and car traffic along highways. From the mathematical point of view, the term hyperbolic net/network indicates a physical problem modelled by hyperbolic conservation/balance laws defined on a collection of 1D domains forming a topological graph. The net/network branches are called edges, and points where the edges meet are called vertices. Coupling conditions at vertices induce complex nonlinear wave interactions. Numerical implementation of these coupling conditions at the discrete level poses significant challenges for the numerical schemes development and mathematical analysis. This chapter presents a review of the numerical methods for hyperbolic problems on networks and nets. The chapter is organized as follows. In Section 2.1 we present the basic ideas behind the modelling that leads to hyperbolic net problems, and we showcase a couple of examples. The first relates to the nets such as vascular stents, bridges, buildings made of frame structures, etc., which can be modelled by the Antman–Cosserat curved rod model. The second example discusses in detail the associated simpler version of the hyperbolic net problem, which is the nonlinear wave equation net. Then in Section 2.2 we provide examples for hyperbolic networks, including vehicular traffic on road networks, irrigation channels, and blood flow. Section 3 illustrates numerical methods for hyperbolic nets and networks. In particular, we talk about finite volume schemes and discontinuous Galerkin methods for hyperbolic nets and networks, together with some applications to ODE-PDE systems arising in vehicular traffic models.

Hyperbolic nets ; Hyperbolic networks ; Numerical methods

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