Abstract
AbstractWe consider kinetic and associated macroscopic equations on networks. A general approach to derive coupling conditions for the macroscopic equations from coupling conditions of the underlying kinetic problem is presented using 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. The procedure is explained for two simplified situations. The linear case is discussed for a linear kinetic BGK-type model leading in the macroscopic limit to a linear hyperbolic problem. The nonlinear situation is investigated for a kinetic relaxation model and an associated macroscopic scalar nonlinear hyperbolic conservation law on a network. 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.KeywordsKinetic and hyperbolic equationsNetworksZero relaxation coupling conditionsHalf-space problems
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