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By R. Ansorge, W. Törnig

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Extra resources for Numerische Losung nichtlinearer partieller Differential- und Integrodifferentialgleichungen

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19) con dato iniziale g1 − g2 e dato laterale h1 − h2 su x = 0. 20) a w si trova R R [u1 (x, t) − u2 (x, t)]2 dx ≤ 0 t (g1 − g2 )2 dx + a 0 (h1 − h2 )2 ds, 0 che mostra come lo scarto quadratico tra le soluzioni sia controllato da quello tra i dati. 19) dipende con continuit` a dal dato iniziale e da quello laterale su x = 0. 20). 3 Traffico su strada 27 di variabili macroscopiche come la densit` a di auto 3 ρ, la loro velocit` a media v e il 4 loro flusso q. Le tre funzioni ρ, v e q (pi` u o meno regolari) sono legate tra loro dalla semplice relazione convettiva q = vρ.

Osservando le caratteristiche, si trova u = 0 se x < 0 e u = 1 se x > t. Nel settore S = {0 < x < t} non passano caratteristiche. Procedendo come nel caso del modello per il traffico, in S definiamo u come un’onda di rarefazione che raccordi con continuit` a i valori 0 e 1. Si perviene in tal modo alla soluzione continua ⎧ x≤0 ⎪ ⎨0 x u (x, t) = t 0< x< t ⎪ ⎩ 1 x ≥ t. Non `e l’unica soluzione generalizzata ˙! Esiste anche un’onda d’urto che `e soluzione, con curva di shock uscente dall’origine. Poich´e u− = 0, u+ = 1, q u− = 0, q u+ = 1 , 2 la condizione di Rankine-Hugoniot d` a s˙ (t) = q(u+ ) − q(u− ) 1 = .

28) La scelta di un eventuale valore di g in x = 0 non `e rilevante. Supponiamo che al tempo t = 0 il semaforo diventi verde. Analizziamo quel che succede. Al verde, il traffico comincia a muoversi: all’inizio, solo le macchine pi` u vicine al semaforo lo superano mentre la maggior parte rimane ferma. Essendo q (ρ) = vm 1 − ρ2ρ , la velocit`a locale dell’onda progressiva `e data m da x0 < 0 −vm q (g (x0 )) = vm x0 > 0, per cui le caratteristiche sono le rette x = −vm t + x0 x = vm t + x0 se x0 < 0 se x0 > 0.

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