This is a bit of a soft question, however, the concept is still bugging me in my studies.
If we consider the following proposition: Solutions to the linear transport equation are constant along characteristic curves.
How can I interpret this geometrically, let alone, in general?
Recall: the linear nonuniform transport equation is as follows
$$U_t + c(x)U_x=0$$
Best Answer
To understand this statement, we need to determine the characteristic curves. For this purpose, we solve $x'(t) = c(x(t))$ with $x(0)=x_0$. The solutions of this differential equation are the curves $t\mapsto x(t)$ along which $U$ is constantly equal to its initial value $U(x_0,0)$. Therefore, the characteristic curves are the paths along which information propagates. Note that
if $c$ is constant, then the characteristics are straight parallel lines with slope $c$ in the $x$-$t$ plane;
if $c$ is piecewise constant, then the characteristics are piecewise linear and parallel;
if $c$ is linear, then the characteristics are exponentials, etc.