Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation

@article{Borsche2018KineticLA,
  title={Kinetic Layers and Coupling Conditions for Macroscopic Equations on Networks I: The Wave Equation},
  author={R. Borsche and A. Klar},
  journal={SIAM J. Sci. Comput.},
  year={2018},
  volume={40}
}
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… Expand
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References

SHOWING 1-10 OF 39 REFERENCES
Kinetic and related macroscopic models for chemotaxis on networks
In this paper we consider kinetic and associated macroscopic models for chemotaxis on a network. Coupling conditions at the nodes of the network for the kinetic problem are presented and used toExpand
Coupling conditions for gas networks governed by the isothermal Euler equations
TLDR
Coupling conditions for gas transport in networks where the governing equations are the isothermal Euler equations are investigated and additional assumptions to obtain a solution near the intersection are introduced. Expand
Half-space kinetic equations with general boundary conditions
TLDR
The main technique is a damping adding-removing procedure, which establishes the well-posedness of linear (or linearized) half-space equations with general boundary conditions and quasi-optimality of the numerical scheme. Expand
A classification of well‐posed kinetic layer problems
In the first part of this paper, we study the half space boundary value problem for the Boltzmann equation with an incoming distribution, obtained when considering the boundary layer arising in theExpand
A Macro-kinetic Hybrid Model for Traffic Flow on Road Networks
TLDR
A new hybrid model for an heterogeneous traffic flow is developed, based on a coupling of the Lighthill — Whitham and Richards macroscopic model and the kinetic model, which reproduces the capacity drop phenomenon at a merge junction without imposing any priority rule. Expand
Damped wave systems on networks: exponential stability and uniform approximations
We consider a damped linear hyperbolic system modeling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a uniqueExpand
A convergent method for linear half-space kinetic equations
We give a unified proof for the well-posedness of a class of linear half-space equations with general incoming data and construct a Galerkin method to numerically resolve this type of equations in aExpand
Nonlinear Boundary Layers of the Boltzmann Equation: I. Existence
Abstract: We study the half-space problem of the nonlinear Boltzmann equation, assigning the Dirichlet data for outgoing particles at the boundary and a Maxwellian as the far field. We will show thatExpand
Kinetic Theory of Evaporation and Condensation : Hydrodynamic Equation and Slip Boundary Condition
The steady behavior of a gas in contact with its condensed phase of arbitrary shape is investigated on the basis of kinetic theory. The Knudsen number of the system (the mean free path of the gasExpand
A fully-discrete-state kinetic theory approach to traffic flow on road networks
TLDR
A new approach to the modeling of vehicular traffic flows on road networks based on kinetic equations is presented, taking advantage of a discrete structure of the space of microscopic states of the vehicles, which is also significant in view of including the intrinsic microscopic granularity of the system in the mesoscopic representation. Expand
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