Method of characteristics

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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.

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[edit] 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 equation (ODE). Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.

[edit] Linear and quasilinear cases

Consider a PDE of the form

\sum_{i=1}^n a_i \frac{\partial u}{\partial x_i}=0.

For this PDE to be linear, the coefficients ai = ai(x1,...,xn) may be functions of the spatial variables. For it to be quasilinear, ai = ai(x1,...,xn,u) 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

(x_1,\dots,x_n,u) = (x_1(s),\dots,x_n(s),u(s))

such that the following system of ODEs is satisfied

\frac{dx_i}{ds} = a_i(x_1,\dots,x_n,u)\qquad\qquad (1)
\frac{du}{ds} = 0.\qquad\qquad (2)

Equations (1) and (2) give the characteristics of the PDE. Equation (2) has the interpretation that the value of a solution u is constant along the characteristics. Note that, if u=u(x1,...,xn) is a solution of the original PDE, then the second equation follows from the first, since

\frac{du}{ds} = \sum_i \frac{\partial u}{\partial x_i} \frac{d x_i}{ds} = \sum_i \frac{\partial u}{\partial x_i} a_i = 0.

Thus the solution is constant along the characteristics. Conversely, any C1 function which is constant along the characteristics is a solution of the original PDE.

[edit] Fully nonlinear case

Consider the partial differential equation

F(x_1,\dots,x_n,u,p_1,\dots,p_n)=0\qquad\qquad (1)

where the variables pi are shorthand for the partial derivatives

p_i = \frac{\partial u}{\partial x_i}.

Let (xi(s),u(s),pi(s)) be a curve in R2n+1. Suppose that u is any solution, and that

u(s) = u(x_1(s),\dots,x_n(s)).

Along a solution, differentiating (1) with respect to s gives

\sum(F_{x_i} + F_u p_i)\dot{x}_i + \sum F_{p_i}\dot{p}_i = 0
\dot{u} - \sum p_i \dot{x}_i = 0
\sum (\dot{x}_i dp_i - \dot{p}_i dx_i)= 0.

(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

\dot{x}_i=\lambda F_{p_i},\quad\dot{p}_i=-\lambda(F_{x_i}+F_up_i),\quad \dot{u}=\lambda\sum p_iF_{p_i}

where λ is a constant. Writing these equations more symmetrically, one obtains the Charpit-Lagrange equations for the characteristic

\frac{\dot{x}_i}{F_{p_i}}=-\frac{\dot{p}_i}{F_{x_i}+F_up_i}=\frac{\dot{u}}{\sum p_iF_{p_i}}

[edit] Example

As an example, consider the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).

a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0\,

where a\, is constant and u\, is a function of x\, and t\,. We want to transform this linear first order PDE into an ODE along the appropriate curve; i.e. something of the form

\frac{d}{ds}u(x(s), t(s)) = F(u, x(s), t(s)) ,

where (x(s),t(s))\, is a characteristic line. First, we find

\frac{d}{ds}u(x(s), t(s)) = \frac{\partial u}{\partial x} \frac{dx}{ds} + \frac{\partial u}{\partial t} \frac{dt}{ds}

by the chain rule. Now, if we set \frac{dx}{ds} = a and \frac{dt}{ds} = 1 we get

a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} \,

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

\frac{d}{ds}u = a \frac{\partial u}{\partial x} + \frac{\partial u}{\partial t} = 0

So, along the characteristic line (x(s), t(s))\,, the original PDE becomes the ODE u_s = F(u, x(s), t(s)) = 0\,. Already we can make a very important observation: along the characteristics the solution is constant. Thus, u(x_s, t_s) = u(x_0, 0)\, where (x_s, t_s)\, and (x_0, 0)\, lie on the same characteristic. But we are not done yet--the exact solution awaits! Now we have three ODEs to solve.

  • \frac{dt}{ds} = 1, letting t(0)=0\, we know t=s\,,
  • \frac{dx}{ds} = a, letting x(0)=x_0\, we know x=as+x_0=at+x_0\,,
  • \frac{du}{ds} = 0, letting u(0)=f(x_0)\, we know u(x(t), t)=f(x_0)=f(x-at)\,.

So, we can conclude that the characteristic lines are straight lines with slope a\,, and the value of u\, remains constant along any characteristic line.


[edit] 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 shockwaves. Intuitively, we can think of each characteristic line implying a solution to u\, along itself. Thus, when two characteristics cross two solutions are implied. This causes shockwaves and the solution to u\, 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.

[edit] External links

[edit] Bibliography

  • Courant, R. and Hilbert, D. (1962). Methods of Mathematical Physics, Volume II. Wiley-Interscience. 
  • L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
  • A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
  • A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9
  • Sarra, Scott The Method of Characteristics with applications to Conservation Laws, Journal of Online Mathematics and its Applications, 2003.
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