An explicit finite volume algorithm for vanishing viscosity solutions on a network

Abstract

<p style='text-indent:20px;'>In [Andreianov, Coclite, Donadello, Discrete Contin. Dyn. Syst. A, 2017], a finite volume scheme was introduced for computing vanishing viscosity solutions on a single-junction network, and convergence to the vanishing viscosity solution was proven. This problem models <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula> incoming and <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula> outgoing roads that meet at a single junction. On each road the vehicle density evolves according to a scalar conservation law, and the requirements for joining the solutions at the junction are defined via the so-called vanishing viscosity germ. The algorithm mentioned above processes the junction in an implicit manner. We propose an explicit version of the algorithm. It differs only in the way that the junction is processed. We prove that the approximations converge to the unique entropy solution of the associated Cauchy problem.</p>

Well-posedness theory for nonlinear scalar conservation laws on networks

“…(This is opposed to the explicit method of Towers [29] where the vertex is modelled as having zero width for any ∆x > 0.) We will use the notational convention that u k,n 0 ≡ u n 0 for all k ∈ I .…”

“…This was done for schemes which are implicit on the nodes in [1, Section 3.2] and [2]. Convergence of a fully explicit scheme for the strictly concave case was shown in [29]. For a general overview over numerical methods for conservation laws on graphs see [6,Section 6].…”

Network models for nonlocal traffic flow

“…Note that for the vanishing viscosity approach no distribution and priority parameters have to be prescribed and the maximum densities on all roads have to be equal. However, an approximate solution can be obtained by the numerical scheme presented in [46].…”

A Central Scheme for Two Coupled Hyperbolic Systems

“…A related approach has been followed in [10], where a kinetic relaxation model governs the coupling conditions. Another alternative is the construction of vanishing viscosity solutions, addressed in the schemes in [46,55], that avoids the use of Lax curves at the expense of formally treating parabolic systems. The system case that is discussed here requires a detailed analysis of the modified coupling condition imposed to the relaxation formulation, which we will consider.…”