By Sandro Salsa, Federico Vegni, Anna Zaretti, Paolo Zunino

This booklet is designed as a complicated undergraduate or a first-year graduate direction for college students from a variety of disciplines like utilized arithmetic, physics, engineering. It has advanced whereas instructing classes on partial differential equations over the past decade on the Politecnico of Milan. the most objective of those classes used to be twofold: at the one hand, to coach the scholars to understand the interaction among idea and modelling in difficulties coming up within the technologies and nevertheless to offer them an exceptional heritage for numerical equipment, corresponding to finite transformations and finite elements.

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**Example text**

In this case we have to weaken the concept of solution and the computation technique. We will come back on these questions later. For the moment, we analyze the method of characteristics in some particularly signiﬁcant cases. 3 The green light problem. Rarefaction waves Suppose that bumper-to-bumper traﬃc is standing at a red light, placed at x = 0, while the road ahead is empty. 32) 0 for x > 0. At time t = 0 the traﬃc light turns green and we want to describe the car ﬂow evolution for t > 0. At the beginning, only the cars nearer to the light start moving while most remain standing.

11) represents a travelling wave, moving with speed v in the positive x−direction. In Fig. e. with speed v = 1. 1 Distributed source Suppose now we take into account the eﬀect of an external distributed source of pollutant along the channel, of intensity f = f (x, t), measured in concentration per unit time. 13) since q = v c, with the initial condition c (x, 0) = g (x) . 14) Again, to compute the value of the solution u at a point (¯ x, t¯) is not diﬃcult. Let x = x0 + vt be the characteristic passing through (¯ x, t¯) and compute u along this characteristic, setting w (t) = c (x0 + vt, t).

Clearly we run the risk of constructing many solutions, each one depending on the way we regularize the initial data, but for the moment we are satisﬁed if we construct at least one solution. 32 2 Scalar Conservation Laws gε = ρ m ⎛ x⎞ gε = ρ m ⎜ 1 − ⎟ ⎝ ε⎠ gε = 0 x =ε 0 Fig. 10. Smoothing of the initial data in the green light problem a) Let us choose as gε the function (Fig. 10) ⎧ x≤0 ⎪ ⎨ ρm x ρ ) 0