Method of characteristics
Encyclopedia
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
.
(ODE). 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 motivation, 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
Suppose that a solution z is known, and consider the surface graph z = z(x,y) in R3. 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 = u(x,y) at every point. In other words, the graph of the solution must be a union of integral curve
s of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.
The equations of the characteristic curve may be expressed invariantly by the Lagrange-Charpit equations
or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):
These are the characteristic equations for the original system.
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, ai 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.
where the variables pi are shorthand for the partial derivatives
Let (xi(s),u(s),pi(s)) be a curve in R2n+1. Suppose that u is any solution, and that
Along a solution, differentiating (1) 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 du-Σpidxi=0.) 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.
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. So to determine the general solution, it is enough to find the characteristics by solving the characteristic system of ODEs:
In this case, the characteristic lines are straight lines with slope
, and the value of 
remains constant along any characteristic line.
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. The zeros of σP, away from the zero section of T∗X, are the characteristics of P. A hypersurface of X defined by the equation F(x) = 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.
One can use the crossings of the characteristics to find shock wave
s. Intuitively, we can think of each characteristic line implying a solution to
along itself. Thus, when two characteristics cross two solutions are implied. This causes shock waves and the solution to 
becomes a multivalued function
. Solving PDEs with this behavior is a very difficult problem and an active area of research.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, 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
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation that, roughly speaking, has a well-posed initial value problem for the first n−1 derivatives. More precisely, the Cauchy problem can be locally solved for arbitrary initial data along...
. 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
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
.
Characteristics of first-order partial differential equations
For a first-order PDE, the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an ordinary differential equationOrdinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
(ODE). 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 motivation, we confine our attention to the case of a function of two independent variables x and y for the moment. Consider a quasilinear
Quasilinear
Quasilinear may refer to:* Quasilinear function, a function that is both quasiconvex and quasiconcave* Quasilinear utility, an economic utility function linear in one argument...
PDE of the form
Suppose that a solution z is known, and consider the surface graph z = z(x,y) in R3. 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 = u(x,y) at every point. In other words, the graph of the solution must be a union of integral curve
Integral curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations...
s of this vector field. These integral curves are called the characteristic curves of the original partial differential equation.
The equations of the characteristic curve may be expressed invariantly by the Lagrange-Charpit equations
or, if a particular parametrization t of the curves is fixed, then these equations may be written as a system of ordinary differential equations for x(t), y(t), z(t):
These are the characteristic equations for the original system.
Linear and quasilinear cases
Consider now a PDE of the formFor this PDE to be linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
, the coefficients ai may be functions of the spatial variables only, and independent of u. For it to be quasilinear, ai 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.
Fully nonlinear case
Consider the partial differential equationwhere the variables pi are shorthand for the partial derivatives
Let (xi(s),u(s),pi(s)) be a curve in R2n+1. Suppose that u is any solution, and that
Along a solution, differentiating (1) with respect to s gives
(The second equation follows from applying the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
to a solution u, and the third follows by taking an exterior derivative
Exterior derivative
In differential geometry, the exterior derivative extends the concept of the differential of a function, which is a 1-form, to differential forms of higher degree. Its current form was invented by Élie Cartan....
of the relation du-Σpidxi=0.) 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
Monge cone
In the mathematical theory of partial differential equations , the Monge cone is a geometrical object associated with a first-order equation. It is named for Gaspard Monge. In two dimensions, letF = 0\qquad\qquad...
of the differential equation should everywhere be tangent to the graph of the solution.
Example
As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).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 findby the chain rule. Now, if we set
and we getwhich 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. So 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 
.
In this case, the characteristic lines are straight lines with slope
, and the value of remains constant along any characteristic line.Characteristics of linear differential operators
Let X be a differentiable manifoldDifferentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
and P a linear differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
of order k. In a local coordinate system xi,
in which α denotes a multi-index. The principal symbol
Symbol of a differential operator
In mathematics, the symbol of a linear differential operator associates to a differential operator a polynomial by, roughly speaking, replacing each partial derivative by a new variable. The symbol of a differential operator has broad applications to Fourier analysis. In particular, in this...
of P, denoted σP, is the function on the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
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
Homogeneous function
In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by a factor, then the result is multiplied by some power of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...
of degree k in the ξ variable. The zeros of σP, away from the zero section of T∗X, are the characteristics of P. A hypersurface of X defined by the equation F(x) = 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 wave
Shock wave
A shock wave is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium or in some cases in the absence of a material medium, through a field such as the electromagnetic field...
s. Intuitively, we can think of each characteristic line implying a solution to
along itself. Thus, when two characteristics cross two solutions are implied. This causes shock waves and the solution to becomes a multivalued functionMultivalued function
In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
. Solving PDEs with this behavior is a very difficult problem and an active area of research.
Characteristics may fail to cover part of the domain of the PDE. This is called a rarefaction
Rarefaction
Rarefaction is the reduction of a medium's density, or the opposite of compression.A natural example of this is as a phase in a sound wave or phonon. Half of a sound wave is made up of the compression of the medium, and the other half is the decompression or rarefaction of the medium.Another...
, and indicates the solution typically exists only in a weak, i.e. integral equation
Integral equation
In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign. There is a close connection between differential and integral equations, and some problems may be formulated either way...
, 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
Finite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
scheme is best for the problem.