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
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
along which the solution can be integrated from some initial data given on a suitable
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
.
Characteristics of first-order partial differential equation
For a first-order PDE (
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
), the method of characteristics discovers curves (called characteristic curves or just characteristics) along which the PDE becomes an
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
(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 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
Suppose that a solution ''z'' is known, and consider the surface graph ''z'' = ''z''(''x'',''y'') 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''(''x'',''y'') 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 curve
In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
Name
Integral curves are known by various other names, depending on the nature and interpr ...
s 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 form
:
For 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 quasilinear,
''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 quasilinear Case
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
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 equation
where the variables ''p''
i are shorthand for the partial derivatives
:
Let (''x''
i(''s''),''u''(''s''),''p''
i(''s'')) 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. The second order partial differential equation is solved with
Charpit method .
Example
As an example, consider the
advection equation
In the field of physics, engineering, and earth sciences, advection is the transport of a substance or quantity by bulk motion of a fluid. The properties of that substance are carried with it. Generally the majority of the advected substance is a ...
(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 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
.
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 manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One m ...
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
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. It may be described also as the dual bundle to the tangent bundle. This m ...
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 ''dx''
''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''(''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
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
s 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
Rarefaction is the reduction of an item's density, the opposite of compression. Like compression, which can travel in waves (sound waves, for instance), rarefaction waves also exist in nature. A common rarefaction wave is the area of low relati ...
, 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
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for th ...
scheme is best for the problem.
See also
*
Method of quantum characteristics
Notes
References
*
*
*
*
* .
*
External links
Prof. Scott Sarra tutorial on Method of Characteristics
{{Numerical PDE
Partial differential equations
Hyperbolic partial differential equations