Why does the Method of Characteristics matter?

Question

I'm learning the method of characteristics from Evans' book on PDE (Chapter 3). From what I understand, a characteristic curve of a PDE is a curve on which the solution $u$ does not vary with respect to time. I.e, for all points $(X(t),t)$, with $X(0) = x$ that $u(X(t),t) = u(x,0)$. But I have a couple questions. First, why do we care about characteristics? Intuitively, it seems like we can construct curves that "cover" all of spacetime, and then piece them together to get a global solution. But how can we apply the method to solve first order equations? And what does it mean when the curves intersect? It seems like the solution loses smoothness. Is this what the whole theory of shocks is about?

Answer 1

The first point about the Method of Characteristics, that may not be evident if you haven't used it "in anger", is that it makes differential equations much easier to solve. Clearly this is a significant advantage.

Secondly, the structures developed in the Method (in particular, the characteristics themselves) are often very informative about the behaviour of solutions even if we can't find a solution explicitly: one can often prove useful general statements about the solution, such as "the solution exists and is smooth until at least this explicitly given time", or "we " even if the solution itself is unknown or not completely specified. Another useful aspect of this is the ability to quickly determine what kinds of initial data are suitable to produce a consistent solution, and where one can expect to determine the solution.

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One can think of the Method as deriving a set of bespoke coordinates that are well-adapted to the PDE, because along a characteristic, the PDE becomes an ODE. This is the key point, and where a lot of the power comes from: because it's now an ODE, this means that the value of the solution at a point is completely determined by the values on the characteristic through the point before that point. (If there is more than one characteristic through a point, we need more information; this is how shocks form: through inconsistent demands from two characteristics passing through a point. Indeed, one often loses more than smoothness: it is quite possible that the solution stops being continuous.). In particular, we can determine the solution uniquely at a point if the initial data curve intersects the characteristic through that point precisely once (less and we know nothing about the value, more and the information is likely to be inconsistent), and the initial data curve is not parallel to the characteristic at the intersection point (this is for technical, Inverse Function Theorem–related reasons.).

The set of characteristics provide a parametrisation of the domain of the solution: the distance along the characteristic in its parametrisation provides one coordinate, while the other is essentially just a label, that is normally chosen judiciously so that the initial data curve is parameterised by the line with the first coordinate equal to zero.

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The simplest example where shocks develop is provided by the Burgers Equation, $$ u_t + uu_x = 0. $$ The characteristic equations on a curve $(T(s),X(s))$ in this case are $$ \frac{dT}{ds} = 1, \qquad \frac{dX}{ds} = u(T(s,r),X(s,r)), \qquad \frac{du}{ds} = 0, $$ so the characteristics depend on the value of $u$. We can nevertheless solve these equations quite easily: $T=s+A(r)$ for some function $r$, and if the initial data is $u(0,x) = \phi(x)$, then $u(T(s,r),X(s,r))=\phi(X(0,r))$. We also then dictate that $T(0,r)=0$ and $X(0,r)=r$ to parametrise the line $T=0$, so $u(s,r)=\phi(0,r)$. But of course then $u$ does not depend on $s$, so we can solve the third equation to find $X(s,r) = r+s\phi(r)$. So we conclude that $$ T(s,r) = s, \qquad X(s,r) = r+s\phi(r), \qquad u(T(s,r),X(s,r)) = u(s,r+s\phi(r)) = \phi(r). $$ Another way of writing an equation for $u$ is to pretend that $u$ is itself an independent variable, so $r=T-uX$, which gives the implicit equation $ u= \phi(X-uT)$, which can be solved by iteration if not explicitly.

When do characteristics cross? Returning to $X(s,r)=r+s\phi(r)$, we can see that there are two characteristics through $(T,X)$ if $r_1+T\phi(r_1) = r_2 + T\phi(r_2)$, i.e. $-1/T = (\phi(r_1)-\phi(r_2))/(r_1-r_2)$, which by the mean value theorem occurs when there is a point $r_3$ between $r_1$ and $r_2$ with $\phi'(r_3) = -1/T$. So characteristics can cross (and shocks occur) whenever $\phi'<0$ somewhere, and not before $T=1/\sup (-\phi')$.

A nice set of notes that build up the method for more and more general first-order PDEs is found here.