Combinatorial Theory. Proc. conf. Rauischholzhausen, 1982 by D. Jungnickel, K. Vedder

By D. Jungnickel, K. Vedder

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E. 1. Another fundamental diagram was considered by Greenberg and was supported by experimental data from the Lincoln tunnel in New York. 4) where v0 is a positive constant. 2. 1 Lighthill-Whitham-Richards Model v PSfrag replacements 49 f vmax vmax ρmax 2 0 ρ ρmax ρmax 2 ρ ρmax Fig. 1. 3). f v PSfrag replacements v0 ρmax e 0 ρmax ρmax e ρ ρmax ρ Fig. 2. 4). A third fundamental diagram is given by the Underwood model, whose velocity function is ρ v(ρ) = vmax e(− ρmax ) . 3. 5. 1) is strictly hyperbolic and the characteristic field is genuinely nonlinear.

14. The interaction between the two waves produces a single wave connecting ul with ur . The variation before t = t¯ due to the interacting waves is given by |ul − um |+|um − ur |, while the variation after t = t¯ due to the wave produced is given by |ul − ur |. Triangular inequality implies that |ul − ur | |ul − um | + |um − ur | and so the proof is finished. The following theorem holds. 6. Let f : R → R be smooth and u ¯ ∈ L1 (R) with bounded variation. 34), defined for every t 0. 39) 0. Proof.

44) For the case i = j, let us indicate by σi,1 and σi,2 the strengths of the interacting waves, then it holds |σi,1 + σi,2 − σi | + k=i |σk | ≤ C|σi,1 ||σi,2 |. 45) One can now fix a parameter δν and split rarefactions in rarefaction fans with shocks of strength at most δν . Also, at each interaction time, one solves 38 2 Conservation Laws exactly the new Riemann problem, eventually splitting the rarefaction waves in rarefaction fans, only if the product of interacting waves is bigger than δν .

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