By Sandro Salsa, Federico M. G. Vegni, Anna Zaretti, Paolo Zunino

This publication is designed as a complicated undergraduate or a first-year graduate direction for college students from quite a few disciplines like utilized arithmetic, physics, engineering. It has developed whereas instructing classes on partial differential equations over the last 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 thought and modelling in difficulties coming up within the technologies and nonetheless to offer them an exceptional history for numerical tools, equivalent to finite changes and finite components.

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**Example text**

There are several possibilities, for instance: Convection (drift). The ﬂux is determined by the water stream only. This case corresponds to a bulk of pollutant that is driven by the stream, without deformation or expansion. Translating into mathematical terms we ﬁnd q (x, t) = vc (x, t) where, we recall, v denotes the stream speed. Diﬀusion. The pollutant expands from higher concentration regions to lower ones. Here we can adopt the so called Fick’s law which reads q (x, t) = −Dcx (x, t) where the constant D depends on the pollutant and has physical dimensions 2 −1 ([D] = [length] × [time] ).

However the model we will present is often in agreement with observations also in this case. 2. No car “sources” or “sinks”. We consider a road section without exit/ entrance gates. 3. The average speed is not constant and depends on the density alone, that is v = v (ρ) . This rather controversial assumption means that at a certain density the speed is uniquely determined and that a density change causes an immediate speed variation. Clearly dv v (ρ) = ≤0 dρ since we expect the speed to decrease as the density increases.

Thus, an appropriate model for their velocity is v˜ (ρ) = v (ρ) − ε ρx ρ which corresponds to q˜ (ρ) = ρv (ρ) − ερx for the ﬂow-rate of cars. Another reason comes from the fact that shocks constructed by the vanishing viscosity method are physical shocks, since they satisfy the entropy inequality. As for the heat equation, in principle we expect to obtain smooth solutions even with discontinuous initial data. On the other hand, the nonlinear term may force the evolution towards a shock wave. 56) connecting two constant states uL and uR , that is, satisfying the conditions lim u (x, t) = uL , lim u (x, t) = uR .