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

John D. Towers

doi:10.3934/nhm.2021021

Article Contents

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

This issue Previous Article Well-posedness theory for nonlinear scalar conservation laws on networks Next Article $ \Gamma $-convergence of quadratic functionals with non uniformly elliptic conductivity matrices

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: April 2021

Revised: July 2021

Published: February 2022

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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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Mathematics Subject Classification: Primary: 90B20; Secondary: 35L65.

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Figure 1. Example 1. Left panel: $ P^0 = 1/5 $. Right panel: $ P^0 $ computed by the fixed point iteration (2.9), or by choosing $ P^0 = 4/5 $. Solid line: $ u_1 $, dashed line: $ u_2 $, dot-dashed line: $ u_3 $. In the left panel a spurious bump in $ u_2 $ is visible, due to a bad choice of $ P^0 $ (which does not affect convergence). In the right panel there is no spurious bump

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