In mathematics, the method of characteristics is a technique for solving partial differential equations.
Typically, it applies to first-order equations, though in general characteristic curves can also be found for hyperbolic and parabolic partial differential equation.
The method is to reduce a partial differential equation (PDE) to a family of ordinary differential equations (ODE) along which the solution can be integrated from some initial data given on a suitable hypersurface.
[1][2] Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.
For the sake of simplicity, we confine our attention to the case of a function of two independent variables x and y for the moment.
Consider a quasilinear PDE of the form[3] Suppose that a solution z is known, and consider the surface graph z = z(x,y) in R3.
In other words, the graph of the solution must be a union of integral curves of this vector field.
These integral curves are called the characteristic curves of the original partial differential equation and follow as the solutions of the characteristic equations:[3] A parametrization invariant form of the Lagrange–Charpit equations is:[5] Consider now a PDE of the form For this PDE to be linear, the coefficients ai may be functions of the spatial variables only, and independent of u.
For it to be quasilinear,[6] ai may also depend on the value of the function, but not on any derivatives.
For a linear or quasilinear PDE, the characteristic curves are given parametrically by such that the following system of ODEs is satisfied Equations (2) and (3) give the characteristics of the PDE.
In the quasilinear case, the use of the method of characteristics is justified by Grönwall's inequality.
Letting capital letters be the solutions to the ODE we find
We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for
It is a straightforward application of Grönwall's Inequality to show that since
Consider the partial differential equation where the variables pi are shorthand for the partial derivatives Let (xi(s),u(s),pi(s)) be a curve in R2n+1.
Manipulating these equations gives 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).
We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form where
Let X be a differentiable manifold and P a linear differential operator of order k. In a local coordinate system xi, in which α denotes a multi-index.
The principal symbol of P, denoted σP, is the function on the cotangent bundle T∗X defined in these local coordinates by where the ξi are the fiber coordinates on the cotangent bundle induced by the coordinate differentials dxi.
Although this is defined using a particular coordinate system, the transformation law relating the ξi and the xi ensures that σP is a well-defined function on the cotangent bundle.
The function σP is homogeneous of degree k in the ξ variable.
One can use the crossings of the characteristics to find shock waves for potential flow in a compressible fluid.
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.
[8] 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.