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

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.

[1]  Y. Sone,et al.  Kinetic Theory of Evaporation and Condensation : Hydrodynamic Equation and Slip Boundary Condition , 1978 .

[2]  François Golse,et al.  Knudsen layers from a computational viewpoint , 1992 .

[3]  Axel Klar,et al.  Well-posedness of Networked Hyperbolic Systems of Balance Laws , 2012, Constrained Optimization and Optimal Control for Partial Differential Equations.

[4]  Michael Herty,et al.  A Macro-kinetic Hybrid Model for Traffic Flow on Road Networks , 2009, Comput. Methods Appl. Math..

[5]  Herbert Egger,et al.  Damped wave systems on networks: exponential stability and uniform approximations , 2016, Numerische Mathematik.

[6]  A. Klar,et al.  THE SCALAR KELLER–SEGEL MODEL ON NETWORKS , 2014 .

[7]  J. Ferziger,et al.  Model Dependence of the Slip Coefficient , 1967 .

[8]  R. Colombo,et al.  On the coupling of systems of hyperbolic conservation laws with ordinary differential equations , 2010 .

[9]  Axel Klar,et al.  Half-moment closure for radiative transfer equations , 2002 .

[10]  Enrique Zuazua,et al.  Stabilization of the Wave Equation on 1-d Networks , 2009, SIAM J. Control. Optim..

[11]  François Golse,et al.  A classification of well‐posed kinetic layer problems , 1988 .

[12]  Tai-Ping Liu,et al.  Boundary Singularity for Thermal Transpiration Problem of the Linearized Boltzmann Equation , 2014 .

[13]  Axel Klar,et al.  Existence of Solutions for Supply Chain Models Based on Partial Differential Equations , 2007, SIAM J. Math. Anal..

[14]  Graziano Guerra,et al.  Balance Laws with Integrable Unbounded Sources , 2008, SIAM J. Math. Anal..

[15]  Qin Li,et al.  A convergent method for linear half-space kinetic equations , 2014, 1408.6630.

[16]  Qin Li,et al.  Half-space kinetic equations with general boundary conditions , 2015, Math. Comput..

[17]  S. K. Loyalka,et al.  Approximate Method in the Kinetic Theory , 1971 .

[18]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[19]  A. Bensoussan,et al.  Boundary Layers and Homogenization of Transport Processes , 1979 .

[20]  Mauro Garavello.  A review of conservation laws on networks , 2010, Networks Heterog. Media.

[21]  Mauro Garavello,et al.  Traffic Flow on a Road Network , 2005, SIAM J. Math. Anal..

[22]  R. Natalini,et al.  A hyperbolic model of chemotaxis on a network: a numerical study , 2012, 1206.2911.

[23]  François Golse,et al.  ANALYSIS OF THE BOUNDARY LAYER EQUATION IN THE KINETIC THEORY OF GASES , 2008 .

[24]  Tong Tang,et al.  Nonlinear Boundary Layers of the Boltzmann Equation , 2002 .

[25]  Michael Herty,et al.  On 2˟2 Conservation Laws at a Junction , 2008, SIAM J. Math. Anal..

[26]  Axel Klar,et al.  Gas flow in pipeline networks , 2006, Networks Heterog. Media.

[27]  F. Golse,et al.  A numerical method for computing asymptotic states and outgoing distributions for kinetic linear half-space problems , 1995 .

[28]  G. Leugering,et al.  On an inverse problem for tree-like networks of elastic strings , 2010 .

[29]  Enrique Zuazua,et al.  Controllability of tree-shaped networks of vibrating strings , 2001 .

[30]  Luisa Fermo,et al.  A fully-discrete-state kinetic theory approach to traffic flow on road networks , 2014, 1406.4257.

[31]  Axel Klar,et al.  Coupling conditions for gas networks governed by the isothermal Euler equations , 2006, Networks Heterog. Media.

[32]  S. Ukai,et al.  Nonlinear Boundary Layers of the Boltzmann Equation: I. Existence , 2003 .

[33]  J. R. Thomas,et al.  Strong evaporation into a half space , 1981 .

[34]  R. E. Marshak,et al.  Note on the Spherical Harmonic Method As Applied to the Milne Problem for a Sphere , 1947 .

[35]  A. Klar,et al.  Kinetic and related macroscopic models for chemotaxis on networks , 2015, 1512.07001.

[36]  Carlo Cercignani,et al.  A variational principle for boundary value problems in kinetic theory , 1969 .

[37]  F. Camilli,et al.  Parabolic models for chemotaxis on weighted networks , 2015, 1511.07279.

[38]  C. Bardos,et al.  DIFFUSION APPROXIMATION AND COMPUTATION OF THE CRITICAL SIZE , 1984 .