# Method of characteristics

In mathematics, the

**method of characteristics**is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equation to a family of ordinary differential equations along which the solution can be integrated from some initial data given on a suitable hypersurface.

## Characteristics of first-order partial differential equation

For a first-order PDE, the method of characteristics discovers curves along which the PDE becomes an ordinary differential equation. 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 PDE of the form

Suppose that a solution

*z*is known, and consider the surface graph

*z*=

*z*in

**R**

^{3}. A normal vector to this surface is given by

As a result, equation is equivalent to the geometrical statement that the vector field

is tangent to the surface

*z*=

*z*at every point, for the dot product of this vector field with the above normal vector is zero. 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 are given by the

*Lagrange-Charpit equations*

A parametrization invariant form of the

*Lagrange-Charpit equations*is:

### Linear and quasilinear cases

Consider now a PDE of the formFor this PDE to be linear, the coefficients

*a*

_{i}may be functions of the spatial variables only, and independent of

*u*. For it to be ,

*a*

_{i}may also depend on the value of the function, but not on any derivatives. The distinction between these two cases is inessential for the discussion here.

For a linear or quasilinear PDE, the characteristic curves are given parametrically by

such that the following system of ODEs is satisfied

Equations and give the characteristics of the PDE.

#### Proof for Quasi-Linear Case

In the Quasi-Linear case, the use of the method of characteristics is justified by Grönwall's Inequality. The above equation may be written asWe 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

Examining, we find, upon differentiating that

which is the same as

We cannot conclude the above is 0 as we would like, since the PDE only guarantees us that this relationship is satisfied for

,,

and we do not yet know that.

However, we can see that

since by the PDE, the last term is 0. This equals

By the triangle inequality, we have

Assuming are at least, we can bound this for small times. Choose a neighborhood around small enough such that are Locally Lipschitz. By continuity, will remain in for small enough. Since, we also have that will be in for small enough by continuity. So, and for. Additionally, for some for by compactness. From this, we find the above is bounded as

for some. It is a straightforward application of Grönwall's Inequality to show that since we have for as long as this inequality holds. We have some interval such that in this interval. Choose the largest such that this is true. Then, by continuity,. Provided the ODE still has a solution in some interval after, we can repeat the argument above to find that in a larger interval. Thus, so long as the ODE has a solution, we have.

### Fully nonlinear case

Consider the partial differential equationwhere the variables

*p*

_{i}are shorthand for the partial derivatives

Let,

*u*,

*p*

_{i}) be a curve in

**R**

^{2n+1}. Suppose that

*u*is any solution, and that

Along a solution, differentiating with respect to

*s*gives

The second equation follows from applying the chain rule to a solution

*u*, and the third follows by taking an exterior derivative of the relation. 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.

For a pedagogical way of deriving the Lagrange-Charpit equations see the chapter 4 at .

## Example

As an example, consider the advection equation.where is constant and is a function of and. We want to transform this linear first-order PDE into an ODE along the appropriate curve; i.e. something of the form

where is a characteristic line. First, we find

by the chain rule. Now, if we set and we get

which is the left hand side of the PDE we started with. Thus

So, along the characteristic line, the original PDE becomes the ODE. That is to say that along the characteristics, the solution is constant. Thus, where and lie on the same characteristic. Therefore, to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs:

- , letting we know,
- , letting we know,
- , letting we know.

## Characteristics of linear differential operators

Let*X*be a differentiable manifold and

*P*a linear differential operator

of order

*k*. In a local coordinate system

*x*

^{i},

in which α denotes a multi-index. The principal symbol of

*P*, denoted σ

_{P}, is the function on the cotangent bundle T

^{∗}

*X*defined in these local coordinates by

where the ξ

_{i}are the fiber coordinates on the cotangent bundle induced by the coordinate differentials d

*x*

^{i}. Although this is defined using a particular coordinate system, the transformation law relating the ξ

_{i}and the

*x*

^{i}ensures that σ

_{P}is a well-defined function on the cotangent bundle.

The function σ

_{P}is homogeneous of degree

*k*in the ξ variable. The zeros of σ

_{P}, away from the zero section of T

^{∗}

*X*, are the characteristics of

*P*. A hypersurface of

*X*defined by the equation

*F*=

*c*is called a characteristic hypersurface at

*x*if

Invariantly, a characteristic hypersurface is a hypersurface whose conormal bundle is in the characteristic set of

*P*.

## 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 waves for potential flow in a compressible fluid. Intuitively, we can think of each characteristic line implying a solution to 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.

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 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.