Editing Method of characteristics (section)

== Qualitative analysis of characteristics ==
Characteristics are also a powerful tool for gaining qualitative insight into a PDE.

One can use the crossings of the characteristics to find [[shock wave]]s for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to <math>u</math> along itself. Thus, when two characteristics cross, the function becomes multi-valued resulting in a non-physical solution. Physically, this contradiction is removed by the formation of a shock wave, a tangential discontinuity or a weak discontinuity and can result in non-potential flow, violating the initial assumptions.<ref>{{citation |first=Lokenath |last=Debnath |authorlink=Lokenath Debnath |title=Nonlinear Partial Differential Equations for Scientists and Engineers |location=Boston |publisher=Birkhäuser |edition=2nd |year=2005 |isbn=0-8176-4323-0 |chapter=Conservation Laws and Shock Waves |pages=251–276 }}</ref>

Characteristics may fail to cover part of the domain of the PDE.  This is called a [[rarefaction]], and indicates the solution typically exists only in a weak, i.e. [[integral equation]], sense.

The direction of the characteristic lines indicates the flow of values through the solution, as the example above demonstrates.  This kind of knowledge is useful when solving PDEs numerically as it can indicate which [[finite difference]] scheme is best for the problem.