Article Contents
Article Contents

# Well-posedness theory for nonlinear scalar conservation laws on networks

• * Corresponding author: Markus Musch

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

• 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.

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

 Citation:

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

Figure 2.  A network with a periodic edge

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

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

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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