The seminal work of Armbruster-Degond-Ringhofer introduced a conservation law model as limit of discrete-event ones. The simplified model of Goettlich-Herty-Klar (GHK) consists of coupled ODE-PDEs for the queues (ODE) and processors (PDE). A complete theory for the GHK model was developed and new models were introduced. The relative numeric is based on Euler-Upwing scheme type.

Books and surveys:

  1. Bressan, S. Canic, M. Garavello, M. Herty, B. Piccoli: Flows on networks: recent results and perspectives, European Mathematical Society Survey, 1 (2014), 47-111. Survey
  2. D’Apice, S. Goettlich, M. Herty, B. Piccoli: Modeling, Simulation and Optimization of Supply Chains. A Continuous Approach, SIAM book series on Mathematical Modeling and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. link

Selected Papers:

  1. D’Apice, R. Manzo, B. Piccoli: Numerical schemes for the optimal input flow of a supply-chain, SIAM Journal on Numerical Analysis, 51 (2013), 2634-2650. pdf
  2. Herty, C. Joerres, B. Piccoli: Existence of solutions to supply chain models based on partial di_erential equations with discontinuous ux function, Journal on Mathematical Analysis and Applications, 401 (2013), 510-517. pdf
  3. D’Apice, R. Manzo, B. Piccoli: Optimal input flows for a Pde-Ode model of supply chains, Communications in Mathematical Sciences, 10 (2012), 1225-1240. pdf
  4. Cutolo, L. Rarita, B. Piccoli: An Upwind-Euler scheme for an ODE-PDE model of supply chains, SIAM Journal on Scientific Computing, 33 (2011), 1669-1688. pdf
  5. Herty, A. Klar, B. Piccoli: Existence of solutions for supply chain models based on partial differential equations, SIAM Journal on Mathematical Analysis, 39 (2007), 160-173. pdf