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

John D. Towers; John D. Towers

doi:10.3934/nhm.2021021

Networks and Heterogeneous Media

2022, Volume 17, Issue 1: 1-13. doi: 10.3934/nhm.2021021

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

John D. Towers

MiraCosta College, 3333 Manchester Avenue, Cardiff-by-the-Sea, CA 92007-1516, USA

Received: 01 April 2021 Revised: 01 July 2021
Primary: 90B20; Secondary: 35L65

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 $ m $ incoming and $ n $ 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.
- Conservation laws,
- network,
- vanishing viscosity,
- finite volume algorithm,
- convergence
Citation: John D. Towers. An explicit finite volume algorithm for vanishing viscosity solutions on a network[J]. Networks and Heterogeneous Media, 2022, 17(1): 1-13. doi: 10.3934/nhm.2021021

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Abstract

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 $ m $ incoming and $ n $ 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.

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