Well-posedness theory for nonlinear scalar conservation laws on networks

Markus Musch; Ulrik Skre Fjordholm; Nils Henrik Risebro

doi:10.3934/nhm.2021025

Article Contents

2022, Volume 17, Issue 1: 101-128. Doi: 10.3934/nhm.2021025

This issue Previous Article Telegraph systems on networks and port-Hamiltonians. Ⅱ. Network realizability Next Article An explicit finite volume algorithm for vanishing viscosity solutions on a network

Well-posedness theory for nonlinear scalar conservation laws on networks

Department of Mathematics, University of Oslo, Postboks 1053, Blindern, 0316 Oslo, Norway

^* Corresponding author: Markus Musch
^* Corresponding author: Markus Musch

Received: July 31, 2021

Revised: September 30, 2021

Published: February 2022

USF was partially supported by the Research Council of Norway project INICE, project no. 301538

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Abstract

We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.

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Mathematics Subject Classification: Primary: 65M12, 35L65; Secondary: 65M08.

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Figure 1. A star shaped network with two ingoing and three outgoing edges

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Figure 2. A network with a periodic edge

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Figure 3. Initial state and state at $ t = 0.7 $ of a {Burgers-type equation} with travelling shock wave which hits the vertex at time $ t = 1-\frac{1}{\sqrt{2}} $. Here, the graph includes a periodic edge

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Figure 4. Initial state at $ t = 0 $ and state at $ t = 0.2 $ of a traffic flow problem with an initial shock at the vertex developing a different elementary wave on each outgoing edge

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Table 1. $ L^1 $ errors and estimated orders of convergence (EOC) for a selection of examples

	Example 7.3		Example 7.5		Example 7.4		Example 7.1		Example 7.2
Grid level	$ L^1 $ error	EOC	$ L^1 $ error	EOC	$ L^1 $ error	EOC	$ L^1 $ error	EOC	$ L^1 $ error	EOC
3	0.10877	-	0.11630	-	0.14459	-	0.07087	-	0.09904	-
4	0.05496	0.98	0.07136	0.70	0.08016	0.85	0.0546	0.38	0.04913	1.01
5	0.03649	0.59	0.04372	0.71	0.04651	0.79	0.03117	0.81	0.02844	0.79
6	0.02629	0.47	0.02255	0.96	0.02711	0.78	0.01903	0.71	0.01627	0.81
7	0.01830	0.52	0.01360	0.73	0.01495	0.86	0.01115	0.77	0.00919	0.82
8	0.01255	0.54	0.00653	1.06	0.00925	0.69	0.00644	0.79	0.00527	0.80
9	0.00883	0.51	0.00325	1.01	0.00480	0.95	0.00330	0.96	0.00268	0.98
10	0.00625	0.50	0.00160	1.02	0.00295	0.70	0.00173	0.93	0.00150	0.84
11	0.00442	0.50	0.00086	0.90	0.00152	0.96	0.00085	1.03	0.00084	0.84
12	0.00312	0.50	0.00040	1.10	0.00081	0.91	0.00042	1.02	0.00047	0.84

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