Kinetic layers and coupling conditions for macroscopic equations on networks I: the wave equation
Provides a systematic derivation of coupling conditions for wave equations on networks, which is important for modeling transport phenomena in networked systems, but the approach is incremental as it extends existing kinetic-to-macroscopic limit methods to networks.
This paper derives coupling conditions for the wave equation on networks from a kinetic BGK model via asymptotic analysis near nodes, using a new approximate method for kinetic half-space problems. Numerical comparisons show good agreement between macroscopic and kinetic solutions for tripod and more complex networks.
We consider kinetic and associated macroscopic equations on networks. The general approach will be explained in this paper for a linear kinetic BGK model and the corresponding limit for small Knudsen number, which is the wave equation. Coupling conditions for the macroscopic equations are derived from the kinetic conditions via an asymptotic analysis near the nodes of the network. This analysis leads to the consideration of a fixpoint problem involving the coupled solutions of kinetic half-space problems. A new approximate method for the solution of kinetic half-space problems is derived and used for the determination of the coupling conditions. Numerical comparisons between the solutions of the macroscopic equation with different coupling conditions and the kinetic solution are presented for the case of tripod and more complicated networks.