Introdution for Transport Equations

The transport equation is given by

(67)\[ \partial_{t}u+c\partial_{x}u=0 \]

Fig. 11 Wave evolution in space and time: a) 2D plots over time; 3D plot in \(u\times x\times t\).

As \(u\) is constant through a characteristic line, if we take advective derivative of \(u\) evolving over time through the characteristic line, we have

\[ \frac{d u[x,t]}{dt}= \frac{d u[ct+x_{0},t]}{dt}=0 \quad \quad \rightarrow \quad \quad \partial_{t}u\frac{dt}{dt}+\partial_{x}u\frac{d(ct+x_{0})}{dt}= \partial_{t}u +c\partial_{x}u=0, \]

recovering a transport equation.

Solving Transport Equation

Method of Characteristics

Exercise 1