Method of characteristics
For a linear or quasilinear PDE, the characteristic curves are given parametrically by
In the quasilinear case, the use of the method of characteristics is justified by Grönwall's inequality. The above equation may be written as
We must distinguish between the solutions to the ODE and the solutions to the PDE, which we do not know are equal a priori. Letting capital letters be the solutions to the ODE we find
where λ is a constant. Writing these equations more symmetrically, one obtains the Lagrange–Charpit equations for the characteristic
Geometrically, the method of characteristics in the fully nonlinear case can be interpreted as requiring that the Monge cone of the differential equation should everywhere be tangent to the graph of the solution.
As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).
Characteristics are also a powerful tool for gaining qualitative insight into a PDE.
The direction of the characteristic lines indicate 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.