TITLE: Explaining the method of characteristics
QUESTION [22 upvotes]: I am learning about solving p.d.e.s by the method of characteristics at the moment. I was given an "algorithm" to solve these problems but I want to know also what is going on, how it works and what it is doing intuitively/physically/graphically. It would be great if someone could either explain it or provide a good link To reference material. Thank you.

REPLY [2 votes]: As an intuitive summary, a surface described by a first order pde inherently has a vector field defined on the surface (see Monge cones). So naturally the vector field provides an associated set of flow lines which cover the surface. Now along a single flow line, as we move down the line, we simply follow the vector field, so the line is described by an ode. The method of characteristics is finding the ode along the flow line.